{"id":9749,"date":"2026-07-25T20:27:00","date_gmt":"2026-07-25T11:27:00","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9749"},"modified":"2026-07-27T15:55:45","modified_gmt":"2026-07-27T06:55:45","slug":"math-abstraction-08-measure-spaces","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/mathematical-abstraction\/math-abstraction-08-measure-spaces\/","title":{"rendered":"\uce21\ub3c4\uacf5\uac04\uacfc \ud655\ub960\uacf5\uac04"},"content":{"rendered":"<p><!-- \uc218\ud559\uc801 \ucd94\uc0c1\ud654 8\ud3b8: \uce21\ub3c4\uc640 \ud655\ub960\uc758 \uacf5\ub9ac\ud654 --><\/p>\n<div class=\"math-abs-series\">\n<p><a href=\"..\/math-abstraction-07-topological-spaces\/\">7\ubd80<\/a>\uc5d0\uc11c \uc911\ub300\ud55c \uc9c8\ubb38 \ud558\ub098\ub97c \ub0a8\uae30\uace0 \ub9c8\ubb34\ub9ac\ud558\uc600\ub2e4. \uc2e4\uc218 \uad6c\uac04 \\([a,\\,b]\\)\uc758 \uae38\uc774 \\(b-a\\)\ub97c \ub2e8\uc21c\ud55c \uc120\ubd84\uc774 \uc544\ub2cc \ubcf5\uc7a1\ud55c \ubd80\ubd84\uc9d1\ud569\uc5d0\uae4c\uc9c0 \ud655\uc7a5\ud560 \uc218 \uc788\uc744\uae4c? \uc120\ud0dd\uacf5\ub9ac\ub97c \uac00\uc815\ud558\uba74 \uc2e4\uc218\uc758 \ube44\uac00\uce21 \ubd80\ubd84\uc9d1\ud569\uc744 \uad6c\uc131\ud560 \uc218 \uc788\ub2e4. \ub530\ub77c\uc11c \uad6c\uac04\uc758 \uae38\uc774\uc640 \uc77c\uce58\ud558\uace0, \ud3c9\ud589\uc774\ub3d9\uc5d0 \ubd88\ubcc0\uc774\uba70, \uac00\uc0b0\uac00\ubc95\uc801\uc778 \uce21\ub3c4\ub97c \uc2e4\uc218\uc758 \ubaa8\ub4e0 \ubd80\ubd84\uc9d1\ud569\uc5d0 \uc815\uc758\ud560 \uc218\ub294 \uc5c6\ub2e4. \uc5ec\uae30\uc11c \ubd88\uac00\ub2a5\ud55c \uac83\uc740 \uc544\ubb34 \uc9d1\ud569\ud568\uc218\ub098 \ub9cc\ub4dc\ub294 \uc77c\uc774 \uc544\ub2c8\ub77c, \uc6b0\ub9ac\uac00 \uae38\uc774\uc5d0 \uae30\ub300\ud558\ub294 \uc774 \uc138 \uc870\uac74\uc744 \ubaa8\ub450 \uc720\uc9c0\ud558\ub294 \uc77c\uc774\ub2e4.<\/p>\n<p>\uc218\ud559\uc790\ub4e4\uc740 \uc774 \ud55c\uacc4\ub97c \ubc1b\uc544\ub4e4\uc774\uace0, \uce21\uc815\ud560 \uc9d1\ud569\uc758 \ubc94\uc704\uc640 \ud06c\uae30\ub97c \uc7ac\ub294 \ud568\uc218\uac00 \ub9cc\uc871\ud574\uc57c \ud560 \uc870\uac74\uc744 \ubd84\ub9ac\ud558\uc5ec \uce21\ub3c4\ub860\uc744 \uc138\uc6e0\ub2e4. \uc774 \ucd94\uc0c1\uc801 \uad6c\uc870\ub294 \ud6d7\ub0a0 \ud655\ub960\ub860\uc5d0\ub3c4 \uacf5\ud1b5\uc758 \uc5c4\ubc00\ud55c \ud615\uc2dd\uc744 \uc81c\uacf5\ud588\ub2e4. <!-- \ub2e4\ub9cc \uce21\ub3c4\ub860\uc801 \uacf5\ub9ac\ud654\uac00 \u201c\ud655\ub960\uc774 \ubcf8\uc9c8\uc801\uc73c\ub85c \ubb34\uc5c7\uc778\uac00\u201d\ub77c\ub294 \ucca0\ud559\uc801 \ub17c\uc7c1\uc744 \ud574\uacb0\ud55c \uac83\uc740 \uc544\ub2c8\ub2e4. --> \uc774 \uae00\uc5d0\uc11c\ub294 \uae38\uc774\uc758 \ud655\uc7a5\uc5d0\uc11c \ucd9c\ubc1c\ud55c \uce21\ub3c4\ub860\uc774 \uc11c\ub85c \ub2e4\ub978 \ud655\ub960 \ud574\uc11d\ub4e4\uc774 \uacf5\uc720\ud560 \uc218 \uc788\ub294 \uc218\ud559\uc801 \uc5b8\uc5b4\uac00 \ub418\ub294 \uacfc\uc815\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n<h4>\ub9ac\ub9cc \uc801\ubd84\uc758 \ud55c\uacc4<\/h4>\n<p>\uc5b4\ub5a4 \uc720\uacc4\ud568\uc218 \\(f:[0,\\,1]\\to\\mathbb R\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \ubd84\ud560 \\(P\\)\ub97c<br \/>\n\\[<br \/>\n0=x_0&lt;x_1&lt;\\cdots&lt;x_n=1<br \/>\n\\]<br \/>\n\uc774\ub77c \ud558\uace0, \uac01 \uc18c\uad6c\uac04 \\([x_{k-1},\\,x_k]\\)\uc5d0\uc11c \\(f\\)\uc758 \uc0c1\ud55c\uacfc \ud558\ud55c\uc744 \uac01\uac01 \\(M_k\\), \\(m_k\\)\ub77c\uace0 \ud558\uc790. \uc774\ub54c<br \/>\n\\[<br \/>\nU(f,\\,P)=\\sum_{k=1}^n M_k(x_k-x_{k-1}),\\qquad<br \/>\nL(f,\\,P)=\\sum_{k=1}^n m_k(x_k-x_{k-1})<br \/>\n\\]<br \/>\n\ub97c \uac01\uac01 <span class=\"defined\">\uc0c1\ud569<\/span>(upper sum)\uacfc <span class=\"defined\">\ud558\ud569<\/span>(lower sum)\uc774\ub77c\uace0 \ud55c\ub2e4. \ubaa8\ub4e0 \ubd84\ud560\uc5d0 \ub300\ud55c \uc0c1\ud569\ub4e4\uc758 \ud558\ud55c\uacfc \ud558\ud569\ub4e4\uc758 \uc0c1\ud55c\uc774 \uc77c\uce58\ud560 \ub54c \\(f\\)\ub97c <span class=\"defined\">\ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5<\/span>(Riemann integrable)\ud558\ub2e4\uace0 \uc815\uc758\ud558\uace0, \uadf8 \uacf5\ud1b5\uac12\uc744 \uc801\ubd84\uac12\uc73c\ub85c \uc0bc\ub294\ub2e4.<\/p>\n<p><!--\n\n\n<p>\uc5c4\ubc00\ud788 \ub9d0\ud558\uba74 \uc704\uc640 \uac19\uc740 \uc815\uc758\ub294 \ub2e4\ub974\ubd80(Gaston Darboux)\uac00 1875\ub144\uc5d0 \uc81c\uc2dc\ud55c \uc815\uc2dd\ud654\uc774\ub2e4. \ub9ac\ub9cc\uc758 \uc6d0\ub798 \uc815\uc758\ub294 \uac01 \uc18c\uad6c\uac04\uc5d0\uc11c \ud45c\ubcf8\uc810\uc744 \ud0dd\ud55c \ud569\uc758 \uadf9\ud55c\uc744 \uc0ac\uc6a9\ud55c\ub2e4. \uadf8\ub7ec\ub098 \uc720\uacc4\ud568\uc218\uc5d0 \ub300\ud574\uc11c \ub450 \uc801\ubd84 \uac00\ub2a5\uc131\uc740 \ub3d9\uce58\uc774\uace0 \uc801\ubd84\uac12\ub3c4 \uac19\uc73c\ubbc0\ub85c, \ud604\ub300 \uad50\uc7ac\uc5d0\uc11c\ub294 \ub2e4\ub974\ubd80 \uc815\uc2dd\ud654\ub97c \ub9ac\ub9cc \uc801\ubd84\uc758 \uc815\uc758\ub85c \ud754\ud788 \uc0ac\uc6a9\ud55c\ub2e4.<a class=\"math-abs-series-ref\" title=\"Darboux, G. (1875). \u201cM\u00e9moire sur les fonctions discontinues.\u201d Annales scientifiques de l\u2019\u00c9cole Normale Sup\u00e9rieure, 2e s\u00e9rie, 4, 57\u2013112.\" href=\"#ref-Darboux1875\">[Darboux1875]<\/a><a class=\"math-abs-series-ref\" title=\"Hawkins, T. (1970). Lebesgue\u2019s Theory of Integration: Its Origins and Development. University of Wisconsin Press.\" href=\"#ref-Hawkins1970\">[Hawkins1970]<\/a><\/p>\n\n\n--><\/p>\n<p>\uc774\uc81c \uad6c\uac04 \\([0,\\,1]\\)\uc5d0\uc11c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub41c \ud568\uc218\ub97c \uc0dd\uac01\ud574\ubcf4\uc790.<br \/>\n\\[<br \/>\nD(x)=<br \/>\n\\begin{cases}<br \/>\n1 &amp; (x\\in\\mathbb Q),\\\\[4pt]<br \/>\n0 &amp; (x\\notin\\mathbb Q).<br \/>\n\\end{cases}<br \/>\n\\]<br \/>\n\uc774\ub97c <span class=\"defined\">\ub514\ub9ac\ud074\ub808 \ud568\uc218<\/span>(Dirichlet function)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>[\ub514\ub9ac\ud074\ub808(Peter Gustav Lejeune Dirichlet)\ub294 1829\ub144 \ud478\ub9ac\uc5d0 \uae09\uc218\uc758 \uc218\ub834\uc744 \ub2e4\ub8ec \ub17c\ubb38 \ub9d0\ubbf8\uc5d0\uc11c, \ub3c5\ub9bd\ubcc0\uc218\uac00 \uc720\ub9ac\uc218\uc77c \ub54c \ud55c \uc0c1\uc218\uc774\uace0 \ubb34\ub9ac\uc218\uc77c \ub54c \ub2e4\ub978 \uc0c1\uc218\uc778 \ud568\uc218\ub97c \uc2e4\uc81c\ub85c \uc81c\uc2dc\ud588\ub2e4. \ubcf8\ubb38\uc758 \\(0\\)\uacfc \\(1\\)\uc744 \ud0dd\ud55c \ud568\uc218\ub294 \uadf8 \uc608\uc758 \ud45c\uc900\ud654\ub41c \ud615\ud0dc\uc774\ub2e4.<!-- \ub2e4\ub9cc \uc774\ub97c \uc5ed\uc0ac\uc0c1 \u201c\ucd5c\ucd08\uc758 \uadf9\ub2e8\uc801 \ubc18\ub840\u201d\ub77c\uace0 \ub2e8\uc815\ud560 \ud544\uc694\ub294 \uc5c6\ub2e4.<a class=\"math-abs-series-ref\" title=\"Lejeune Dirichlet, G. (1829). \u201cSur la convergence des s\u00e9ries trigonom\u00e9triques qui servent \u00e0 repr\u00e9senter une fonction arbitraire entre des limites donn\u00e9es.\u201d Journal f\u00fcr die reine und angewandte Mathematik, 4, 157\u2013169.\" href=\"#ref-Dirichlet1829\">[Dirichlet1829]<\/a> -->]<\/p>\n<p>\uc720\ub9ac\uc218 \uc9d1\ud569\uacfc \ubb34\ub9ac\uc218 \uc9d1\ud569\uc740 \uac01\uac01 \uc2e4\uc218\uc5d0\uc11c \uc870\ubc00\ud558\ub2e4. \ub530\ub77c\uc11c \uae38\uc774\uac00 \uc591\uc218\uc778 \ubaa8\ub4e0 \uc18c\uad6c\uac04\uc5d0\ub294 \uc720\ub9ac\uc218\uc640 \ubb34\ub9ac\uc218\uac00 \ubaa8\ub450 \ub4e4\uc5b4 \uc788\ub2e4. \uc5b4\ub5a4 \ubd84\ud560 \\(P\\)\ub97c \ud0dd\ud558\ub354\ub77c\ub3c4 \uac01 \uc18c\uad6c\uac04\uc5d0\uc11c \\(M_k=1\\), \\(m_k=0\\)\uc774\ubbc0\ub85c<br \/>\n\\[<br \/>\nU(D,\\,P)=1,\\qquad L(D,\\,P)=0<br \/>\n\\]<br \/>\n\uc774\ub2e4. \ubd84\ud560\uc744 \uc544\ubb34\ub9ac \uc798\uac8c \ub9cc\ub4e4\uc5b4\ub3c4 \uc774 \uac04\uaca9\uc740 \uc904\uc5b4\ub4e4\uc9c0 \uc54a\ub294\ub2e4. \ub530\ub77c\uc11c \ub514\ub9ac\ud074\ub808 \ud568\uc218\ub294 \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\uc9c0 \uc54a\ub2e4.<\/p>\n<p>\ub514\ub9ac\ud074\ub808 \ud568\uc218\ub294 \\([0,\\,1]\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \ubd88\uc5f0\uc18d\uc774\ub2e4.<\/p>\n<p>[\uc784\uc758\uc758 \uc810 \\(x_0\\in[0,1]\\)\uacfc \uc784\uc758\uc758 \\(\\delta>0\\)\uc744 \uc7a1\uc544\ub3c4 \\(x_0\\)\uc758 \uc815\uc758\uc5ed \uc548 \\(\\delta\\)-\uadfc\ubc29\uc5d0\ub294 \uc720\ub9ac\uc218\uc640 \ubb34\ub9ac\uc218\uac00 \ubaa8\ub450 \uc874\uc7ac\ud55c\ub2e4. \ub530\ub77c\uc11c \\(x_0\\)\uc73c\ub85c \uc218\ub834\ud558\uba74\uc11c \ud568\uc218\uac12\uc774 \uac01\uac01 \\(1\\)\uacfc \\(0\\)\uc778 \ub450 \uc218\uc5f4\uc744 \uc7a1\uc744 \uc218 \uc788\ub2e4. \ud568\uc218\uac12\uc774 \ud558\ub098\uc758 \uadf9\ud55c\uc73c\ub85c \uc218\ub834\ud560 \uc218 \uc5c6\uc73c\ubbc0\ub85c \\(D\\)\ub294 \\(x_0\\)\uc5d0\uc11c \ubd88\uc5f0\uc18d\uc774\uace0, \\(x_0\\)\uac00 \uc784\uc758\uc600\uc73c\ubbc0\ub85c \ubaa8\ub4e0 \uc810\uc5d0\uc11c \ubd88\uc5f0\uc18d\uc774\ub2e4.]<\/p>\n<p>\ubd88\uc5f0\uc18d\uc810\uacfc \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\uc131\uc758 \uad00\uacc4\ub294 \uc774\ubbf8 \ub9ac\ub9cc\uc758 \uc9c4\ub3d9 \uc870\uac74\uc5d0 \ub4e4\uc5b4 \uc788\uc5c8\uc73c\uba70, \ub974\ubca0\uadf8\ub294 \uc0c8\ub85c\uc6b4 \uce21\ub3c4 \uac1c\ub150\uc744 \uc774\uc6a9\ud574 \uc774\ub97c \uc624\ub298\ub0a0\uc758 \uac04\uacb0\ud55c \ud615\ud0dc\ub85c \ud45c\ud604\ud588\ub2e4.<\/p>\n<p>[\uc624\ub298\ub0a0 <span class=\"defined\">\ub974\ubca0\uadf8 \ud310\uc815\ubc95<\/span>(Lebesgue criterion)\uc774\ub77c\uace0 \ubd80\ub974\ub294 \uc815\ub9ac\uc5d0 \ub530\ub974\uba74, \uc720\uacc4\ud568\uc218 \\(f:[a,b]\\to\\mathbb R\\)\uac00 \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \ubd88\uc5f0\uc18d\uc810\ub4e4\uc758 \uc9d1\ud569\uc774 \ub974\ubca0\uadf8 \uce21\ub3c4 \\(0\\)\uc778 \uac83\uc774\ub2e4. \ub974\ubca0\uadf8\uc758 1902\ub144 \ud559\uc704\ub17c\ubb38\uc740 \ub9ac\ub9cc\uc758 \uc9c4\ub3d9 \uc870\uac74\uc744 \uce21\ub3c4\ub860\uc801 \uc5b8\uc5b4\uc640 \uc5f0\uacb0\ud588\ub2e4. \ub514\ub9ac\ud074\ub808 \ud568\uc218\uc758 \ubd88\uc5f0\uc18d\uc810 \uc9d1\ud569\uc740 \\([0,1]\\) \uc804\uccb4\uc774\uace0 \uadf8 \uce21\ub3c4\ub294 \\(1\\)\uc774\ubbc0\ub85c, \uc774 \ud310\uc815\ubc95\uc73c\ub85c\ub3c4 \ub9ac\ub9cc \uc801\ubd84 \ubd88\uac00\ub2a5\ud558\ub2e4\ub294 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4.<a class=\"math-abs-series-ref\" title=\"Lebesgue, H. (1902). \u201cInt\u00e9grale, Longueur, Aire.\u201d Annali di Matematica Pura ed Applicata, Serie III, 7, 231\u2013359. \uacf5\uac1c \uc2a4\uce94: Internet Archive.\" href=\"#ref-Lebesgue1902\">[Lebesgue1902]<\/a>]<\/p>\n<p><!-- \ub514\ub9ac\ud074\ub808 \ud568\uc218\uac00 \ub974\ubca0\uadf8 \uc774\ub860\uc744 \uc9c1\uc811 \ucd09\ubc1c\ud588\ub2e4\uace0 \ub9d0\ud558\uae30\ub294 \uc5b4\ub835\ub2e4. -->\ub514\ub9ac\ud074\ub808\uc758 \uc608\ub294 1829\ub144 \ud478\ub9ac\uc5d0 \uae09\uc218\uc758 \ub9e5\ub77d\uc5d0\uc11c \ub4f1\uc7a5\ud588\uace0, \ub974\ubca0\uadf8\ub294 \uae38\uc774\u00b7\ub113\uc774\uc758 \uc77c\ubc18\ud654\uc640 \ub9ac\ub9cc \uc801\ubd84\uc758 \uc81c\ud55c\uc744 \ud3ec\ud568\ud55c \uc5ec\ub7ec \ubb38\uc81c\ub97c \ub2e4\ub8e8\uc5c8\ub2e4. \ub2e4\ub9cc \uc774 \ud568\uc218\ub294 \ud604\ub300\uc801 \uad00\uc810\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84\uc758 \uc81c\ud55c\uacfc \ub974\ubca0\uadf8 \uc801\ubd84\uc758 \ud655\uc7a5\uc131\uc744 \uc120\uba85\ud558\uac8c \ube44\uad50\ud574 \uc8fc\ub294 \uc608\uc774\ub2e4.<\/p>\n<h4>\ub974\ubca0\uadf8, \uae38\uc774\ub97c \uce21\ub3c4\ub85c \ud655\uc7a5\ud558\ub2e4<\/h4>\n<p>\ubcf5\uc7a1\ud55c \uc9d1\ud569\uc758 \ud06c\uae30\ub97c \uc815\uc758\ud558\ub294 \ubb38\uc81c\ub294 \ub974\ubca0\uadf8\uc5d0\uac8c\uc11c \uac11\uc790\uae30 \uc2dc\uc791\ub41c \uac83\uc774 \uc544\ub2c8\ub2e4. \ud398\uc544\ub178\uc640 \uc870\ub974\ub2f9\uc740 \uc720\ud55c \uac1c\uc758 \uad6c\uac04\uc774\ub098 \uc9c1\uc0ac\uac01\ud615\uc744 \uc774\uc6a9\ud55c \ub0b4\uc6a9\uc744 \uc5f0\uad6c\ud588\uace0, \ubcf4\ub810\uc740 \uac00\uc0b0 \uac1c\uc758 \uad6c\uac04\uc744 \uc0ac\uc6a9\ud558\ub294 \uce21\ub3c4\ub97c \ubc1c\uc804\uc2dc\ucf30\ub2e4. \ub974\ubca0\uadf8\ub294 \uc774 \uc120\ud589 \uc791\uc5c5\uc744 \ubc14\ud0d5\uc73c\ub85c 1902\ub144 \ud559\uc704\ub17c\ubb38 \u300e\uc801\ubd84, \uae38\uc774, \ub113\uc774\u300f(<i>Int\u00e9grale, Longueur, Aire<\/i>)\uc5d0\uc11c \uce21\ub3c4\uc640 \uc801\ubd84\uc744 \ud06c\uac8c \ud655\uc7a5\ud588\ub2e4.<a class=\"math-abs-series-ref\" title=\"Lebesgue, H. (1902). \u201cInt\u00e9grale, Longueur, Aire.\u201d Annali di Matematica Pura ed Applicata, Serie III, 7, 231\u2013359. \uacf5\uac1c \uc2a4\uce94: Internet Archive.\" href=\"#ref-Lebesgue1902\">[Lebesgue1902]<\/a><a class=\"math-abs-series-ref\" title=\"Hawkins, T. (1970). Lebesgue\u2019s Theory of Integration: Its Origins and Development. University of Wisconsin Press.\" href=\"#ref-Hawkins1970\">[Hawkins1970]<\/a><\/p>\n<p>\ud604\ub300\uc801\uc778 \uc815\uc2dd\ud654\ub85c, \uc2e4\uc9c1\uc120\uc758 \uc784\uc758\uc758 \ubd80\ubd84\uc9d1\ud569 \\(A\\subseteq\\mathbb R\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nm^*(A)=\\inf\\left\\{<br \/>\n\\sum_{n=1}^{\\infty}(b_n-a_n):<br \/>\nA\\subseteq\\bigcup_{n=1}^{\\infty}(a_n,b_n)<br \/>\n\\right\\}<br \/>\n\\]<br \/>\n\ub85c \uc815\uc758\ud55c \uac12\uc744 <span class=\"defined\">\ub974\ubca0\uadf8 \uc678\uce21\ub3c4<\/span>(Lebesgue outer measure)\ub77c\uace0 \ud55c\ub2e4. \uc989 \\(A\\)\ub97c \ub36e\ub294 \uac00\uc0b0 \uac1c \uc5f4\ub9b0\uad6c\uac04\uc758 \uae38\uc774 \ud569\uc744 \ubaa8\ub450 \uc0dd\uac01\ud558\uace0 \uadf8 \ud558\ud55c\uc744 \ucde8\ud55c\ub2e4. \uc774 \uac12\uc740 \ubaa8\ub4e0 \ubd80\ubd84\uc9d1\ud569\uc5d0 \ub300\ud574 \\([0,\\infty]\\)\uc758 \uac12\uc744 \uac16\ub294\ub2e4. \uc678\uce21\ub3c4\ub294 \ub2e8\uc870\uc131\uacfc \uac00\uc0b0\uc900\uac00\ubc95\uc131\uc744 \ub9cc\uc871\ud558\uc9c0\ub9cc, \ubaa8\ub4e0 \uc9d1\ud569\uc5d0\uc11c \uac00\uc0b0\uac00\ubc95\uc801\uc778 \u201c\ucc38\ub41c \uae38\uc774\u201d\uac00 \ub418\ub294 \uac83\uc740 \uc544\ub2c8\ub2e4.<a class=\"math-abs-series-ref\" title=\"Lebesgue, H. (1902). \u201cInt\u00e9grale, Longueur, Aire.\u201d Annali di Matematica Pura ed Applicata, Serie III, 7, 231\u2013359. Internet Archive.\" href=\"#ref-Lebesgue1902\">[Lebesgue1902]<\/a><\/p>\n<p>\ub2eb\ud78c\uad6c\uac04 \\([a,b]\\)\uc758 \uc678\uce21\ub3c4\ub294 \uae30\uc874\uc758 \uae38\uc774 \\(b-a\\)\uc640 \uc77c\uce58\ud55c\ub2e4. \uc2e4\uc81c\ub85c \uc784\uc758\uc758 \\(\\varepsilon>0\\)\uc5d0 \ub300\ud558\uc5ec \uc5f4\ub9b0\uad6c\uac04<br \/>\n\\[<br \/>\n(a-\\varepsilon\/2,\\,b+\\varepsilon\/2)<br \/>\n\\]<br \/>\n\uac00 \\([a,b]\\)\ub97c \ub36e\uace0 \uadf8 \uae38\uc774\ub294 \\(b-a+\\varepsilon\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(m^*([a,b])\\le b-a+\\varepsilon\\)\uc774\uace0, \\(\\varepsilon\\)\uc774 \uc784\uc758\uc774\ubbc0\ub85c \\(m^*([a,b])\\le b-a\\)\uc774\ub2e4.<\/p>\n<p>[\ubc18\ub300 \ubd80\ub4f1\uc2dd\uc5d0\ub294 \ud558\uc774\ub124\u2013\ubcf4\ub810 \uc815\ub9ac\ub97c \uc0ac\uc6a9\ud560 \uc218 \uc788\ub2e4. \\([a,b]\\)\uc758 \uc784\uc758\uc758 \uac00\uc0b0 \uc5f4\ub9b0\uad6c\uac04 \ub36e\uac1c\uc5d0\uc11c \uc720\ud55c \ubd80\ubd84\ub36e\uac1c\ub97c \ubf51\uc73c\uba74, \uadf8 \uc720\ud55c \uac1c \uad6c\uac04\uc758 \uae38\uc774 \ud569\uc740 \uc801\uc5b4\ub3c4 \\(b-a\\)\uc774\ub2e4. \ub530\ub77c\uc11c \uc6d0\ub798 \ub36e\uac1c\uc758 \uc804\uccb4 \uae38\uc774 \ud569\ub3c4 \\(b-a\\) \uc774\uc0c1\uc774\uace0, \ubaa8\ub4e0 \ub36e\uac1c\uc5d0 \ub300\ud55c \ud558\ud55c \uc5ed\uc2dc \\(b-a\\) \uc774\uc0c1\uc774\ub2e4. \ub450 \ubd80\ub4f1\uc2dd\uc744 \ud569\uce58\uba74 \\(m^*([a,b])=b-a\\)\uc774\ub2e4.]<\/p>\n<p>\uac00\uc0b0\uc9d1\ud569\uc758 \uc678\uce21\ub3c4\ub294 \ud56d\uc0c1 \\(0\\)\uc774\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \\(\\mathbb Q\\cap[0,1]\\)\uc744<br \/>\n\\[<br \/>\nq_1,q_2,q_3,\\ldots<br \/>\n\\]<br \/>\n\ub85c \ub098\uc5f4\ud558\uace0 \\(\\varepsilon>0\\)\uc744 \uc7a1\uc790. \uac01 \\(q_n\\)\uc744 \uae38\uc774\uac00 \\(\\varepsilon\/2^n\\)\uc778 \uc5f4\ub9b0\uad6c\uac04\uc73c\ub85c \ub36e\uc73c\uba74, \uc774 \uad6c\uac04\ub4e4\uc758 \uae38\uc774 \ud569\uc740 \ub4f1\ube44\uae09\uc218<br \/>\n\\[<br \/>\n\\sum_{n=1}^{\\infty}\\frac{\\varepsilon}{2^n}=\\varepsilon<br \/>\n\\]<br \/>\n\uac00 \ub41c\ub2e4. \ub530\ub77c\uc11c<br \/>\n\\[<br \/>\nm^*(\\mathbb Q\\cap[0,1])\\le\\varepsilon<br \/>\n\\]<br \/>\n\uc774\uace0, \\(\\varepsilon\\)\uc744 \uc784\uc758\ub85c \uc791\uac8c \ud0dd\ud560 \uc218 \uc788\uc73c\ubbc0\ub85c \\(m^*(\\mathbb Q\\cap[0,1])=0\\)\uc774\ub2e4. \uc720\ub9ac\uc218 \uc9d1\ud569\uc740 \uc870\ubc00\ud558\uc9c0\ub9cc \uae38\uc774\ub294 \ucc28\uc9c0\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n<p><!--\n\n\n<p>[\uc5ec\uae30\uc11c\ub294 \ub450 \uc9d1\ud569\uc744 \uad6c\ubcc4\ud574\uc57c \ud55c\ub2e4. \ub514\ub9ac\ud074\ub808 \ud568\uc218\uac00 \\(1\\)\uc778 \uc810\ub4e4\uc758 \uc9d1\ud569 \\(\\mathbb Q\\cap[0,1]\\)\uc740 \uce21\ub3c4 \\(0\\)\uc774\ubbc0\ub85c \\(D=0\\)\uc740 \uac70\uc758 \ubaa8\ub4e0 \uacf3\uc5d0\uc11c \uc131\ub9bd\ud558\uace0, \\(D\\)\uc758 \ub974\ubca0\uadf8 \uc801\ubd84\uac12\uc740 \\(0\\)\uc774\ub2e4. \uadf8\ub7ec\ub098 \\(D\\)\uc758 \ubd88\uc5f0\uc18d\uc810 \uc9d1\ud569\uc740 \\(\\mathbb Q\\cap[0,1]\\)\uc774 \uc544\ub2c8\ub77c \\([0,1]\\) \uc804\uccb4\uc774\ub2e4. \ub974\ubca0\uadf8 \ud310\uc815\ubc95\uc740 \ud568\uc218\uac00 \\(0\\)\uc774 \uc544\ub2cc \uc810\ub4e4\uc758 \uc9d1\ud569\uc774 \uc544\ub2c8\ub77c \ubd88\uc5f0\uc18d\uc810\ub4e4\uc758 \uc9d1\ud569\uc744 \uac80\uc0ac\ud558\ubbc0\ub85c, \\(D\\)\ub294 \uc5ec\uc804\ud788 \ub9ac\ub9cc \uc801\ubd84 \ubd88\uac00\ub2a5\ud558\ub2e4.]<\/p>\n\n\n--><\/p>\n<p>\uc678\uce21\ub3c4\uac00 \ubaa8\ub4e0 \ubd80\ubd84\uc9d1\ud569\uc5d0 \uc678\uce21\uac12\uc744 \ubc30\uc815\ud55c\ub2e4\uace0 \ud574\uc11c \ubb38\uc81c\uac00 \ub05d\ub098\uc9c0\ub294 \uc54a\ub294\ub2e4. \uc11c\ub85c\uc18c\uc778 \uac00\uc0b0 \uac1c\uc758 \uc9d1\ud569\uc744 \ud569\ucce4\uc744 \ub54c \uc804\uccb4\uc758 \uae38\uc774\uac00 \uac01 \uc870\uac01 \uae38\uc774\uc758 \ud569\uacfc \uc77c\uce58\ud558\ub294 <span class=\"defined\">\uac00\uc0b0\uac00\ubc95\uc131<\/span>(countable additivity)\uc740 \uce21\ub3c4\uc758 \ud575\uc2ec \uc870\uac74\uc774\ub2e4. \ub974\ubca0\uadf8 \uc678\uce21\ub3c4\ub294 \uc784\uc758\uc758 \uc9d1\ud569\uc5f4\uc5d0 \ub300\ud574\uc11c \uac00\uc0b0\uc900\uac00\ubc95\uc131<br \/>\n\\[<br \/>\nm^*\\left(\\bigcup_{n=1}^{\\infty}A_n\\right)<br \/>\n\\le\\sum_{n=1}^{\\infty}m^*(A_n)<br \/>\n\\]<br \/>\n\uc744 \ub9cc\uc871\ud558\uc9c0\ub9cc, \ubaa8\ub4e0 \uc11c\ub85c\uc18c \ubd80\ubd84\uc9d1\ud569\uc5f4\uc5d0\uc11c \ub4f1\ud638\ub97c \ubcf4\uc7a5\ud558\uc9c0\ub294 \uc54a\ub294\ub2e4.<\/p>\n<h4>\ube44\ud0c8\ub9ac\uc758 \ube44\uac00\uce21\uc9d1\ud569<\/h4>\n<p>1905\ub144, \uc774\ud0c8\ub9ac\uc544 \uc218\ud559\uc790 \uc8fc\uc138\ud398 \ube44\ud0c8\ub9ac(Giuseppe Vitali)\ub294 \uc120\ud0dd\uacf5\ub9ac\ub97c \uc774\uc6a9\ud558\uc5ec \ub974\ubca0\uadf8 \ube44\uac00\uce21\uc9d1\ud569\uc758 \uc608\ub97c \uc81c\uc2dc\ud588\ub2e4.<a class=\"math-abs-series-ref\" title=\"Vitali, G. (1905). Sul problema della misura dei gruppi di punti di una retta. Bologna: Tipografia Gamberini e Parmeggiani. Google Books.\" href=\"#ref-Vitali1905\">[Vitali1905]<\/a> \uc989, \uad6c\uac04\uc758 \uae38\uc774\ub97c \uc5f0\uc7a5\ud558\ub294 \ud3c9\ud589\uc774\ub3d9\uc5d0 \ub300\ud558\uc5ec \ubd88\ubcc0\uc774\uace0 \uac00\uc0b0\uac00\ubc95\uc801\uc778 \uce21\ub3c4\ub97c \uba71\uc9d1\ud569 \\(\\mathcal{P} (\\mathbb{R})\\) \uc804\uccb4\uc5d0\uc11c \uc815\uc758\ud558\ub294 \uac83\uc740 \ubd88\uac00\ub2a5\ud558\ub2e4.<\/p>\n<p>\ubc18\uc5f4\ub9b0\uad6c\uac04 \\([0,1)\\)\uc758 \ub450 \uc810 \\(x,y\\) \uc0ac\uc774\uc5d0<br \/>\n\\[<br \/>\nx\\sim y\\quad\\Longleftrightarrow\\quad x-y\\in\\mathbb Q<br \/>\n\\]<br \/>\n\ub77c\ub294 \uad00\uacc4\ub97c \uc815\uc758\ud558\uc790. \uc774 \uad00\uacc4\ub294 \ubc18\uc0ac\uc131, \ub300\uce6d\uc131, \ucd94\uc774\uc131\uc744 \ub9cc\uc871\ud558\ubbc0\ub85c \ub3d9\uce58\uad00\uacc4\uc774\ub2e4. \ub530\ub77c\uc11c \\([0,1)\\)\uc740 \uc11c\ub85c\uc18c\uc778 \ub3d9\uce58\ub958\ub4e4\ub85c \ubd84\ud560\ub41c\ub2e4. \uc120\ud0dd\uacf5\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \uac01 \ub3d9\uce58\ub958\uc5d0\uc11c \ub300\ud45c\uc6d0 \ud558\ub098\uc529\uc744 \uace0\ub974\uace0, \uadf8 \ub300\ud45c\uc6d0\ub4e4\uc758 \uc9d1\ud569\uc744 \\(V\\subset[0,1)\\)\ub77c \ud558\uc790.<\/p>\n<p>\uac01 \\(q\\in\\mathbb Q\\cap[0,1)\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nT_q(v)=v+q\\pmod 1,\\qquad<br \/>\nV_q=T_q(V)<br \/>\n\\]<br \/>\n\ub85c \uc815\uc758\ud558\uc790.<\/p>\n<p>[\uc5ec\uae30\uc11c \\(v+q\\pmod1\\)\uc740 \uc2e4\uc9c1\uc120 \uc704\uc758 \ubcf4\ud1b5 \ud3c9\ud589\uc774\ub3d9\uc774 \uc544\ub2c8\ub77c, \\([0,1)\\)\uc758 \uc591 \ub05d\uc744 \ubd99\uc5ec \ub9cc\ub4e0 \uc6d0 \\(\\mathbb R\/\\mathbb Z\\) \uc704\uc758 \ud68c\uc804\uc774\ub2e4. \uad6c\uccb4\uc801\uc73c\ub85c \\(v+q&lt;1\\)\uc774\uba74 \\(T_q(v)=v+q\\)\uc774\uace0, \\(v+q\\ge1\\)\uc774\uba74 \\(T_q(v)=v+q-1\\)\uc774\ub2e4.]<\/p>\n<p>\uc9d1\ud569\ub4e4 \\(V_q\\)\ub294 \uc11c\ub85c\uc18c\uc774\uace0, \uc774\ub4e4\uc758 \ud569\uc9d1\ud569\uc740 \\([0,1)\\) \uc804\uccb4\uc774\ub2e4.<\/p>\n<ul>\n<li>\uba3c\uc800 \uc11c\ub85c\uc18c\uc784\uc744 \ubcf4\uc790. \\(T_{q_1}(v_1)=T_{q_2}(v_2)\\)\ub77c\uace0 \uac00\uc815\ud558\uba74 \uc5b4\ub5a4 \\(k\\in\\mathbb Z\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nv_1+q_1=v_2+q_2+k<br \/>\n\\]<br \/>\n\uc774\ub2e4. \ub530\ub77c\uc11c \\(v_1-v_2=q_2-q_1+k\\in\\mathbb Q\\)\uc774\ubbc0\ub85c \\(v_1\\sim v_2\\)\uc774\ub2e4. \\(V\\)\ub294 \uac01 \ub3d9\uce58\ub958\uc5d0\uc11c \ub300\ud45c\uc6d0 \ud558\ub098\ub9cc \uace8\ub790\uc73c\ubbc0\ub85c \\(v_1=v_2\\)\uc774\ub2e4. \uc774\uc5b4\uc11c \\(q_1-q_2\\in\\mathbb Z\\)\uc774\uace0 \\(q_1,q_2\\in[0,1)\\)\uc774\ubbc0\ub85c \\(q_1=q_2\\)\uc774\ub2e4.<\/li>\n<li>\ub2e4\uc74c\uc73c\ub85c \ud569\uc9d1\ud569\uc744 \ubcf4\uc790. \\(x\\in[0,1)\\)\ub97c \uc7a1\uace0 \\(x\\)\uac00 \uc18d\ud55c \ub3d9\uce58\ub958\uc758 \ub300\ud45c\uc6d0\uc744 \\(v\\in V\\)\ub77c \ud558\uc790. \uadf8\ub7ec\uba74 \\(r=x-v\\in\\mathbb Q\\cap(-1,1)\\)\uc774\ub2e4. \\(r\\ge0\\)\uc774\uba74 \\(q=r\\), \\(r&lt;0\\)\uc774\uba74 \\(q=r+1\\)\ub85c \ub450\uc790. \uadf8\ub7ec\uba74 \\(q\\in\\mathbb Q\\cap[0,1)\\)\uc774\uace0 \\(x=T_q(v)\\)\uc774\ub2e4. \ub530\ub77c\uc11c<br \/>\n\\[<br \/>\n[0,1)=\\bigsqcup_{q\\in\\mathbb Q\\cap[0,1)}V_q<br \/>\n\\]<br \/>\n\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\uc774\uc81c \\(V\\)\uac00 \ub974\ubca0\uadf8 \uac00\uce21\uc774\ub77c\uace0 \uac00\uc815\ud558\uace0 \\(m(V)=c\\)\ub77c \ud558\uc790. \\(V\\subset[0,1)\\)\uc774\ubbc0\ub85c \\(0\\le c\\le1\\)\uc774\ub2e4.<\/p>\n<p>[\ub974\ubca0\uadf8 \uce21\ub3c4\uc758 \ubcf4\ud1b5 \ud3c9\ud589\uc774\ub3d9 \ubd88\ubcc0\uc131\ub9cc \uc801\uace0 \uc21c\ud658\uc774\ub3d9\uc758 \ubd88\ubcc0\uc131\uc744 \uace7\ubc14\ub85c \uacb0\ub860 \ub0b4\ub9ac\uba74 \ud55c \ub2e8\uacc4\uac00 \ube60\uc9c4\ub2e4. \\(0\\le q&lt;1\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nA_q=V\\cap[0,1-q),\\qquad B_q=V\\cap[1-q,1)<br \/>\n\\]<br \/>\n\ub85c \ub450\uba74 \\(V=A_q\\sqcup B_q\\)\uc774\uace0<br \/>\n\\[<br \/>\nT_q(V)=(A_q+q)\\sqcup(B_q+q-1)<br \/>\n\\]<br \/>\n\uc774\ub2e4. \\(V\\)\uac00 \uac00\uce21\uc774\ub77c\ub294 \uac00\uc815 \uc544\ub798 \ub450 \uc870\uac01\ub3c4 \uac00\uce21\uc774\ub2e4. \ud3c9\ud589\uc774\ub3d9 \ubd88\ubcc0\uc131\uacfc \uc720\ud55c\uac00\ubc95\uc131\uc744 \uc801\uc6a9\ud558\uba74<br \/>\n\\[<br \/>\nm(T_q(V))=m(A_q)+m(B_q)=m(V)=c<br \/>\n\\]<br \/>\n\ub97c \uc5bb\ub294\ub2e4. \ub530\ub77c\uc11c \ubaa8\ub4e0 \\(V_q\\)\uc758 \uce21\ub3c4\ub294 \\(c\\)\uc774\ub2e4.<a class=\"math-abs-series-ref\" title=\"Vitali, G. (1905). Sul problema della misura dei gruppi di punti di una retta. Bologna: Tipografia Gamberini e Parmeggiani.\" href=\"#ref-Vitali1905\">[Vitali1905]<\/a>]<\/p>\n<p>\uac00\uc0b0\uac00\ubc95\uc131\uc744 \uc801\uc6a9\ud558\uba74<br \/>\n\\[<br \/>\n1=m([0,1))<br \/>\n=\\sum_{q\\in\\mathbb Q\\cap[0,1)}m(V_q)<br \/>\n=\\sum_{q\\in\\mathbb Q\\cap[0,1)}c<br \/>\n\\]<br \/>\n\uac00 \ub418\uc5b4\uc57c \ud55c\ub2e4. \uadf8\ub7ec\ub098 \\(c=0\\)\uc774\uba74 \uc6b0\ubcc0\uc740 \\(0\\)\uc774\uace0, \\(c>0\\)\uc774\uba74 \uc6b0\ubcc0\uc740 \\(\\infty\\)\uc774\ub2e4. \uc5b4\ub290 \uacbd\uc6b0\uc5d0\ub3c4 \\(1\\)\uc774 \ub420 \uc218 \uc5c6\uc73c\ubbc0\ub85c \ubaa8\uc21c\uc774\ub2e4. \ub530\ub77c\uc11c \\(V\\)\ub294 \ub974\ubca0\uadf8 \uac00\uce21\uc9d1\ud569\uc774 \uc544\ub2c8\ub2e4.<\/p>\n<p><!-- \uc774\ub294 \\(V\\)\uc5d0 \uc5b4\ub5a4 \uc885\ub958\uc758 \uc9d1\ud569\ud568\uc22b\uac12\ub3c4 \ubd80\uc5ec\ud560 \uc218 \uc5c6\ub2e4\ub294 \ub73b\uc774 \uc544\ub2c8\ub2e4. \uad6c\uac04\uc758 \uae38\uc774\ub97c \ubcf4\uc874\ud558\uace0 \ud3c9\ud589\uc774\ub3d9\uc5d0 \ubd88\ubcc0\uc774\uba70 \uac00\uc0b0\uac00\ubc95\uc801\uc778 \uce21\ub3c4\uc758 \uc815\uc758\uc5ed\uc5d0 \\(V\\)\uc640 \uadf8 \uc720\ub9ac \uc21c\ud658\uc774\ub3d9\ub4e4\uc744 \ubaa8\ub450 \ud3ec\ud568\ud560 \uc218 \uc5c6\ub2e4\ub294 \ub73b\uc774\ub2e4. -->\ub974\ubca0\uadf8\ub294 1902\ub144\uc5d0 \uac00\uce21\uc9d1\ud569\uc758 \ubc94\uc704\ub97c \uad6c\ubcc4\ud588\uace0, \ube44\ud0c8\ub9ac\uc758 1905\ub144 \uad6c\uc131\uc740 \uc65c \uadf8 \ubc94\uc704\ub97c \ubaa8\ub4e0 \ubd80\ubd84\uc9d1\ud569\uc73c\ub85c \ub113\ud790 \uc218 \uc5c6\ub294\uc9c0\ub97c \uba85\ud655\ud558\uac8c \ubcf4\uc5ec\uc8fc\uc5c8\ub2e4. <!-- \uadf8\ub7ec\ubbc0\ub85c \ube44\ud0c8\ub9ac\uc758 \ubc18\ub840\uac00 \ub974\ubca0\uadf8 \uce21\ub3c4\ub97c \ud0c4\uc0dd\uc2dc\ucf30\ub2e4\uace0 \uc11c\uc220\ud558\uba74 \uc5ed\uc0ac\uc801 \uc21c\uc11c\uac00 \ub4a4\ubc14\ub010\ub2e4. --><\/p>\n<p>[\uce74\ub77c\ud14c\uc624\ub3c4\ub9ac(Constantin Carath\u00e9odory)\ub294 1914\ub144 \uc678\uce21\ub3c4\ub85c\ubd80\ud130 \uac00\uce21\uc9d1\ud569\uc744 \uace8\ub77c\ub0b4\ub294 \uc77c\ubc18\uc801\uc778 \ud310\uc815\ubc95\uc744 \ubc1c\ud45c\ud588\ub2e4.<a class=\"math-abs-series-ref\" title=\"Carath\u00e9odory, C. (1914). \u201c\u00dcber das lineare Ma\u00df von Punktmengen\u2014eine Verallgemeinerung des L\u00e4ngenbegriffs.\u201d Nachrichten von der Gesellschaft der Wissenschaften zu G\u00f6ttingen, Mathematisch-Physikalische Klasse, 404\u2013426. EuDML.\" href=\"#ref-Caratheodory1914\">[Caratheodory1914]<\/a> \uc9d1\ud569 \\(X\\) \uc704\uc758 \uc678\uce21\ub3c4 \\(\\mu^*\\)\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \\(E\\subseteq X\\)\uac00 \ubaa8\ub4e0 \\(A\\subseteq X\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n\\mu^*(A)=\\mu^*(A\\cap E)+\\mu^*(A\\setminus E)<br \/>\n\\]<br \/>\n\ub97c \ub9cc\uc871\ud558\uba74 \\(E\\)\ub97c \uce74\ub77c\ud14c\uc624\ub3c4\ub9ac \uac00\uce21\uc9d1\ud569\uc774\ub77c\uace0 \ud55c\ub2e4. \uc774\ub7ec\ud55c \uc9d1\ud569\ub4e4\uc740 \uc2dc\uadf8\ub9c8\ub300\uc218\ub97c \uc774\ub8e8\uace0, \uadf8 \uc2dc\uadf8\ub9c8\ub300\uc218\uc5d0 \uc81c\ud55c\ud55c \\(\\mu^*\\)\ub294 \uc644\ube44 \uce21\ub3c4\uac00 \ub41c\ub2e4. \ud2b9\ud788 \\(\\mu^*=m^*\\)\uac00 \ub974\ubca0\uadf8 \uc678\uce21\ub3c4\uc774\uba74 \uc774 \ud310\uc815\ubc95\uc73c\ub85c \uc5bb\uc740 \uc9d1\ud569\ub4e4\uc774 \ub974\ubca0\uadf8 \uac00\uce21\uc9d1\ud569\uc774\uace0, \\(m^*\\)\uc758 \uc81c\ud55c\uc774 \ub974\ubca0\uadf8 \uce21\ub3c4\uc774\ub2e4.]<\/p>\n<h4>\ucd94\uc0c1 \uce21\ub3c4\uacf5\uac04<\/h4>\n<p>\ube44\ud0c8\ub9ac \uc9d1\ud569\uc758 \uacb0\ub860\uc740 \uad6c\uac04\uc758 \ud1b5\uc0c1\uc801\uc778 \uae38\uc774\ub97c \ubcf4\uc874\ud558\uace0 \ud3c9\ud589\uc774\ub3d9\uc5d0 \ubd88\ubcc0\uc778 \uac00\uc0b0\uac00\ubc95\uc801 \uce21\ub3c4\uc758 \uc815\uc758\uc5ed\uc744 \uc2e4\uc9c1\uc120\uc758 \uba71\uc9d1\ud569 \uc804\uccb4\ub85c \uc0bc\uc744 \uc218 \uc5c6\uc74c\uc744 \ubcf4\uc5ec \uc900\ub2e4. \ub530\ub77c\uc11c \ub974\ubca0\uadf8 \uae38\uc774\ub97c \ub2e4\ub8f0 \ub54c\uc5d0\ub294 \uc5ec\uc9d1\ud569\uacfc \uac00\uc0b0\ud569\uc9d1\ud569 \uac19\uc740 \uae30\ubcf8 \uc5f0\uc0b0\uc5d0 \ub2eb\ud78c \ubd80\ubd84\uc9d1\ud569\uc871\uc744 \uba3c\uc800 \uc815\ud558\uace0, \uadf8 \uc9d1\ud569\uc871 \uc704\uc5d0\uc11c \uac00\uc0b0\uac00\ubc95\uc801\uc778 \uc9d1\ud569\ud568\uc218\ub97c \uc815\uc758\ud55c\ub2e4. <!-- \uc2dc\uadf8\ub9c8\ub300\uc218\uac00 \uadf8 \uc790\uccb4\ub85c \ub17c\ub9ac\uc801 \ubaa8\uc21c\uc744 \uc790\ub3d9\uc73c\ub85c \ub9c9\uc544 \uc8fc\ub294 \uac83\uc740 \uc544\ub2c8\ub2e4. \uc2dc\uadf8\ub9c8\ub300\uc218\uc640 \uadf8 \uc704\uc758 \uce21\ub3c4\uac00 \ud568\uaed8 \uac01\uac01\uc758 \uacf5\ub9ac\ub97c \ub9cc\uc871\ud574\uc57c \ud55c\ub2e4. --><\/p>\n<div class=\"box theorem\">\n<p><span class=\"theorem\">\uc815\uc758: \ucd94\uc0c1 \uce21\ub3c4\uacf5\uac04<\/span><br \/>\n\uc9d1\ud569 \\(X\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc871 \\(\\mathcal F\\)\uac00 \ub2e4\uc74c \uc870\uac74\uc744 \ub9cc\uc871\ud55c\ub2e4\uace0 \ud558\uc790.<\/p>\n<ul>\n<li>\\(X\\in\\mathcal F\\)\uc774\ub2e4.<\/li>\n<li>\\(A\\in\\mathcal F\\)\uc774\uba74 \\(X\\setminus A\\in\\mathcal F\\)\uc774\ub2e4.<\/li>\n<li>\uac00\uc0b0\uc5f4 \\(A_1,A_2,\\ldots\\in\\mathcal F\\)\uc774 \uc8fc\uc5b4\uc9c0\uba74<br \/>\n\\[<br \/>\n\\bigcup_{n=1}^{\\infty}A_n\\in\\mathcal F<br \/>\n\\]<br \/>\n\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\uc774\ub54c \\(\\mathcal F\\)\ub97c \\(X\\) \uc704\uc758 <span class=\"defined\">\uc2dc\uadf8\ub9c8\ub300\uc218<\/span>(\\(\\sigma\\)-algebra)\ub77c\uace0 \ud558\uace0, \\(\\mathcal F\\)\uc758 \uc6d0\uc18c\ub97c <span class=\"defined\">\uac00\uce21\uc9d1\ud569<\/span>(measurable set)\uc774\ub77c\uace0 \ud55c\ub2e4.<\/p>\n<p>[\uc2dc\uadf8\ub9c8\ub300\uc218\uc758 \uc870\uac74\uc740 \uc55e\uc120 \uae00\uc5d0\uc11c \uc704\uc0c1(topology)\uc744 \uc815\uc758\ud560 \ub54c \uc694\uad6c\ud55c \uc870\uac74\uacfc \ud615\ud0dc\uc801\uc73c\ub85c \ub2ee\uc558\ub2e4. \ub450 \uad6c\uc870 \ubaa8\ub450 \uc9d1\ud569 \uc5f0\uc0b0\uc5d0 \ub300\ud55c \ub2eb\ud798\uc744 \uc694\uad6c\ud55c\ub2e4. \uadf8\ub7ec\ub098 \uc704\uc0c1\uc740 \uc784\uc758\uc758 \ud569\uc9d1\ud569\uacfc \uc720\ud55c \uad50\uc9d1\ud569\uc5d0 \ub2eb\ud600 \uc788\ub294 \ubc18\uba74, \uc2dc\uadf8\ub9c8\ub300\uc218\ub294 \uac00\uc0b0\ud569\uc9d1\ud569\uacfc \uc5ec\uc9d1\ud569\uc5d0 \ub2eb\ud600 \uc788\uc5b4\uc57c \ud55c\ub2e4. \ubaa8\ub4e0 \uc5f4\ub9b0\uc9d1\ud569\uc758 \uc5ec\uc9d1\ud569\uc774 \ub2e4\uc2dc \uc5f4\ub9b4 \ud544\uc694\ub294 \uc5c6\uc73c\ubbc0\ub85c \ub450 \uad6c\uc870\ub294 \uc11c\ub85c \uac19\uc9c0 \uc54a\ub2e4.]<\/p>\n<p>\ud568\uc218 \\(\\mu:\\mathcal F\\to[0,\\infty]\\)\uac00 \ub2e4\uc74c \uc870\uac74\uc744 \ub9cc\uc871\ud55c\ub2e4\uace0 \ud558\uc790.<\/p>\n<ul>\n<li>\\(\\mu(\\varnothing)=0\\)\uc774\ub2e4.<\/li>\n<li>\uc11c\ub85c\uc18c\uc778 \uac00\uc0b0\uc5f4 \\(A_1,A_2,\\ldots\\in\\mathcal F\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n\\mu\\left(\\bigcup_{n=1}^{\\infty}A_n\\right)<br \/>\n=\\sum_{n=1}^{\\infty}\\mu(A_n)<br \/>\n\\]<br \/>\n\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\uc774\ub54c \\(\\mu\\)\ub97c \\((X,\\mathcal F)\\) \uc704\uc758 <span class=\"defined\">\uce21\ub3c4<\/span>(measure)\ub77c\uace0 \ud558\uace0, \uc21c\uc11c\uc0bc\uc911\ud56d \\((X,\\mathcal F,\\mu)\\)\ub97c <span class=\"defined\">\uce21\ub3c4\uacf5\uac04<\/span>(measure space)\uc774\ub77c\uace0 \ud55c\ub2e4.<\/p>\n<\/div>\n<p>\uc774 \uacf5\ub9ac\uacc4\ub294 \uc11c\ub85c \ub2e4\ub978 \u201c\ud06c\uae30\u201d\ub97c \ud55c \uc5b8\uc5b4\ub85c \ub2e4\ub8e8\uac8c \ud55c\ub2e4.<\/p>\n<ul>\n<li>\uc2e4\uc9c1\uc120\uc5d0\uc11c \ub974\ubca0\uadf8 \uac00\uce21\uc9d1\ud569\ub4e4\uc758 \uc2dc\uadf8\ub9c8\ub300\uc218\uc5d0 \ub974\ubca0\uadf8 \uc678\uce21\ub3c4\ub97c \uc81c\ud55c\ud558\uba74 <span class=\"defined\">\ub974\ubca0\uadf8 \uce21\ub3c4<\/span>(Lebesgue measure)\ub97c \uc5bb\ub294\ub2e4. \uc774 \uce21\ub3c4\ub294 \uad6c\uac04\uc5d0 \ud1b5\uc0c1\uc801\uc778 \uae38\uc774\ub97c \ubd80\uc5ec\ud55c\ub2e4.<\/li>\n<li>\uac19\uc740 \uad6c\uc131\uc744 \\(\\mathbb R^2\\)\uc640 \\(\\mathbb R^3\\)\ub85c \ud655\uc7a5\ud558\uba74 \ub113\uc774\uc640 \ubd80\ud53c\ub97c \uc77c\ubc18\ud654\ud55c \ub974\ubca0\uadf8 \uce21\ub3c4\ub97c \uc5bb\ub294\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \uc9d1\ud569 \\(X\\)\uc5d0\uc11c \\(\\mathcal F=\\mathcal P(X)\\)\ub85c \ub450\uace0<br \/>\n\\[<br \/>\n\\mu(A)=<br \/>\n\\begin{cases}<br \/>\n|A| &amp; (A\\text{\uac00 \uc720\ud55c\uc9d1\ud569\uc77c \ub54c}),\\\\[4pt]<br \/>\n\\infty &amp; (A\\text{\uac00 \ubb34\ud55c\uc9d1\ud569\uc77c \ub54c})<br \/>\n\\end{cases}<br \/>\n\\]<br \/>\n\ub85c \uc815\uc758\ud558\uba74 <span class=\"defined\">\uc148\uce21\ub3c4<\/span>(counting measure)\uac00 \ub41c\ub2e4.<\/li>\n<\/ul>\n<p>\uc774 \uce21\ub3c4\uacf5\uac04\ub4e4\uc740 \uac19\uc740 \uacf5\ub9ac\ub97c \ub9cc\uc871\ud558\ubbc0\ub85c \uce21\ub3c4 \uacf5\ub9ac\ub9cc\uc73c\ub85c \uc99d\uba85\ub41c \uc77c\ubc18 \uc815\ub9ac\ub97c \ubaa8\ub450 \uacf5\uc720\ud55c\ub2e4. <!-- \uadf8\ub7ec\ub098 \uac19\uc740 \uacf5\ub9ac\ub97c \ub9cc\uc871\ud55c\ub2e4\uace0 \ud574\uc11c \uacf5\uac04\ub4e4\uc774 \uc11c\ub85c \ub3d9\ud615\uc774\uac70\ub098 \u201c\ub3d9\ub4f1\ud55c \ub300\uc0c1\u201d\uc774 \ub418\ub294 \uac83\uc740 \uc544\ub2c8\ub2e4. \ub2e8, \uacf5\ub9ac\ud654\uac00 \ud1b5\ud569\ud558\ub294 \uac83\uc740 \ubaa8\ub4e0 \uad6c\uccb4\uc801 \ucc28\uc774\uac00 \uc544\ub2c8\ub77c, \uadf8 \uacf5\ub9ac\ub4e4\ub85c\ubd80\ud130 \ub530\ub77c\uc624\ub294 \ucd94\ub860\uc758 \ud615\uc2dd\uc774\ub2e4. --><\/p>\n<p>\uce21\ub3c4\ub294 \ud2b9\ud788 \ub2e8\uc870\ub86d\uac8c \uc99d\uac00\ud558\ub294 \uc9d1\ud569\uc5f4\uc758 \ud569\uc9d1\ud569\uacfc \uc798 \ub9de\ub294\ub2e4. \uc774 \uacbd\uc6b0\uc5d0\ub294 \ud569\uc9d1\ud569\uc744 \ucde8\ud55c \ub4a4 \uce21\uc815\ud55c \uac12\uacfc \uac01 \ub2e8\uacc4\uc758 \uce21\ub3c4\ub4e4\uc758 \uadf9\ud55c\uc774 \uc77c\uce58\ud55c\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac: \uce21\ub3c4\uc758 \uc544\ub798\ub85c\ubd80\ud130\uc758 \uc5f0\uc18d\uc131.<\/span><br \/>\n\uce21\ub3c4\uacf5\uac04 \\((X,\\mathcal F,\\mu)\\)\uc5d0\uc11c<br \/>\n\\[<br \/>\nA_1\\subseteq A_2\\subseteq\\cdots<br \/>\n\\]<br \/>\n\uc778 \uc99d\uac00\uc9d1\ud569\uc5f4\uc774 \uc8fc\uc5b4\uc9c0\uba74<br \/>\n\\[<br \/>\n\\mu\\left(\\bigcup_{n=1}^{\\infty}A_n\\right)<br \/>\n=\\lim_{n\\to\\infty}\\mu(A_n)<br \/>\n\\]<br \/>\n\uc774\ub2e4. \uc774 \uac12\uc740 \\(\\infty\\)\uc77c \uc218\ub3c4 \uc788\ub2e4.\n<\/p>\n<\/div>\n<p>\uc99d\uba85\uc744 \uc704\ud574 \\(B_1=A_1\\)\ub85c \ub450\uace0, \\(n\\ge2\\)\uc77c \ub54c<br \/>\n\\[<br \/>\nB_n=A_n\\setminus A_{n-1}<br \/>\n\\]<br \/>\n\ub85c \ub450\uc790.<br \/>\n[\\(B_n=A_n\\setminus A_{n-1}=A_n\\cap(X\\setminus A_{n-1})\\)\uc774\ub2e4. \uc2dc\uadf8\ub9c8\ub300\uc218\ub294 \uc5ec\uc9d1\ud569\uacfc \uc720\ud55c \uad50\uc9d1\ud569\uc5d0 \ub2eb\ud600 \uc788\uc73c\ubbc0\ub85c \\(B_n\\in\\mathcal F\\)\uc774\ub2e4. \uc720\ud55c \uad50\uc9d1\ud569\uc5d0 \ub300\ud55c \ub2eb\ud798\uc740 \ub4dc\ubaa8\ub974\uac04 \ubc95\uce59\uc744 \uc774\uc6a9\ud574 \uc5ec\uc9d1\ud569\uacfc \uac00\uc0b0\ud569\uc9d1\ud569 \uacf5\ub9ac\uc5d0\uc11c \uc5bb\ub294\ub2e4.]<\/p>\n<p>\uc870\uac01\ub4e4 \\(B_1,B_2,\\ldots\\)\uc740 \uc11c\ub85c\uc18c\uc774\ub2e4. \uc2e4\uc81c\ub85c \\(m&lt;n\\)\uc774\uba74 \\(B_m\\subseteq A_m\\subseteq A_{n-1}\\)\uc778 \ubc18\uba74 \\(B_n\\cap A_{n-1}=\\varnothing\\)\uc774\ub2e4. \ub610\ud55c \uc784\uc758\uc758 \\(N\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n\\bigcup_{n=1}^{N}B_n=A_N<br \/>\n\\]<br \/>\n\uc774\ub2e4.<br \/>\n[\uac00\uc7a5 \uc548\ucabd\uc758 \\(A_1\\)\uc5d0\uc11c \uc2dc\uc791\ud574 \uac01 \ub2e8\uacc4\uc5d0\uc11c \uc0c8\ub85c \ucd94\uac00\ub41c \ubd80\ubd84 \\(A_n\\setminus A_{n-1}\\)\ub9cc \ubd99\uc778 \uac83\uc774\ubbc0\ub85c \\(N\\)\ub2e8\uacc4\uc5d0\uc11c\ub294 \uc815\ud655\ud788 \\(A_N\\)\uc774 \ubcf5\uc6d0\ub41c\ub2e4. \uc5c4\ubc00\ud558\uac8c\ub294 \\(N\\)\uc5d0 \ub300\ud55c \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc73c\ub85c \uc704 \ub4f1\uc2dd\uc744 \uc99d\uba85\ud560 \uc218 \uc788\ub2e4.]<\/p>\n<p>\ub530\ub77c\uc11c<br \/>\n\\[<br \/>\n\\bigcup_{n=1}^{\\infty}B_n<br \/>\n=\\bigcup_{n=1}^{\\infty}A_n<br \/>\n\\]<br \/>\n\uc774\ub2e4.<br \/>\n[\uc67c\ucabd \ud569\uc9d1\ud569\uc5d0 \uc18d\ud558\ub294 \uc6d0\uc18c\ub294 \uc5b4\ub5a4 \\(B_n\\subseteq A_n\\)\uc5d0 \uc18d\ud558\ubbc0\ub85c \uc624\ub978\ucabd\uc5d0\ub3c4 \uc18d\ud55c\ub2e4. \ubc18\ub300\ub85c \uc624\ub978\ucabd \ud569\uc9d1\ud569\uc5d0 \uc18d\ud558\ub294 \uc6d0\uc18c \\(x\\)\uc5d0 \ub300\ud574\uc11c\ub294 \\(x\\in A_n\\)\uc774 \ub418\ub294 \ucd5c\uc18c\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \\(n=1\\)\uc774\uba74 \\(x\\in B_1\\)\uc774\uace0, \\(n\\ge2\\)\uc774\uba74 \\(x\\in A_n\\setminus A_{n-1}=B_n\\)\uc774\ub2e4. \ub530\ub77c\uc11c \ub450 \ud569\uc9d1\ud569\uc740 \uac19\ub2e4.]<\/p>\n<p>\uac00\uc0b0\uac00\ubc95\uc131\uc744 \uc801\uc6a9\ud558\uba74<br \/>\n\\[<br \/>\n\\mu\\left(\\bigcup_{n=1}^{\\infty}A_n\\right)<br \/>\n=\\sum_{n=1}^{\\infty}\\mu(B_n)<br \/>\n=\\lim_{N\\to\\infty}\\sum_{n=1}^{N}\\mu(B_n)<br \/>\n\\]<br \/>\n\uc774\ub2e4.<br \/>\n[\uc720\ud55c\ud569\uc5d0 \ub300\ud55c \uac00\ubc95\uc131\ub3c4 \uac00\uc0b0\uac00\ubc95\uc131\uc5d0\uc11c \uc5bb\uc5b4\uc9c4\ub2e4. \uc11c\ub85c\uc18c\uc778 \\(B_1,\\ldots,B_N\\) \ub4a4\uc5d0 \\(B_{N+1}=B_{N+2}=\\cdots=\\varnothing\\)\uc744 \ubd99\uc774\uba74, \uacf5\uc9d1\ud569\uc758 \uce21\ub3c4\uac00 \\(0\\)\uc774\ubbc0\ub85c \uac00\uc0b0\uac00\ubc95\uc131 \uacf5\ub9ac\ub97c \uadf8\ub300\ub85c \uc801\uc6a9\ud560 \uc218 \uc788\ub2e4.]<\/p>\n<p>\uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[<br \/>\n\\sum_{n=1}^{N}\\mu(B_n)<br \/>\n=\\mu\\left(\\bigcup_{n=1}^{N}B_n\\right)<br \/>\n=\\mu(A_N)<br \/>\n\\]<br \/>\n\uc774\uace0, \ub450 \uc2dd\uc744 \uacb0\ud569\ud558\uba74<br \/>\n\\[<br \/>\n\\mu\\left(\\bigcup_{n=1}^{\\infty}A_n\\right)<br \/>\n=\\lim_{N\\to\\infty}\\mu(A_N)<br \/>\n\\]<br \/>\n\uc744 \uc5bb\ub294\ub2e4.<span class=\"qed\"><\/span><\/p>\n<p>\uc774 \uacb0\uacfc\ub294 \uc99d\uac00\uc9d1\ud569\uc5f4\uc5d0 \ud55c\uc815\ub41c \u201c\uc544\ub798\ub85c\ubd80\ud130\uc758 \uc5f0\uc18d\uc131\u201d\uc774\ub2e4. \uac10\uc18c\uc9d1\ud569\uc5f4 \\(A_1\\supseteq A_2\\supseteq\\cdots\\)\uc5d0\uc11c\ub3c4 \uc720\uc0ac\ud55c \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud558\uc9c0\ub9cc, \uadf8 \uacbd\uc6b0\uc5d0\ub294 \uc77c\ubc18\uc801\uc73c\ub85c \\(\\mu(A_1)&lt;\\infty\\)\ub77c\ub294 \uc870\uac74\uc774 \ud544\uc694\ud558\ub2e4. <!-- \ub530\ub77c\uc11c \u201c\uadf9\ud55c\uacfc \uce21\ub3c4 \uae30\ud638\ub97c \uc5b8\uc81c\ub098 \uc790\uc720\ub86d\uac8c \ub9de\ubc14\uafc0 \uc218 \uc788\ub2e4\u201d\ub77c\uace0 \uc77c\ubc18\ud654\ud574\uc11c\ub294 \uc548 \ub41c\ub2e4. --><\/p>\n<h4>\ucf5c\ubaa8\uace0\ub85c\ud504, \ud655\ub960\uc5d0 \uce21\ub3c4\ub860\uc801 \ud615\uc2dd\uc744 \uc8fc\ub2e4<\/h4>\n<p>\uce21\ub3c4\ub860\uc758 \ubc1c\uc804\uc740 \ud655\ub960\ub860\uc5d0 \uacf5\ud1b5\uc758 \uc5c4\ubc00\ud55c \uc218\ud559\uc801 \ud615\uc2dd\uc744 \uc81c\uacf5\ud588\ub2e4. \uc774 \ud615\uc2dd\uc774 \ud655\ub960\uc758 \ucca0\ud559\uc801 \uc758\ubbf8\uc5d0 \uad00\ud55c \ub17c\uc7c1\uc744 \ub05d\ub0b8 \uac83\uc740 \uc544\ub2c8\uc9c0\ub9cc, \uc11c\ub85c \ub2e4\ub978 \ud574\uc11d\uc744 \ub530\ub974\ub294 \uc5f0\uad6c\uc790\ub4e4\uc774 \uac19\uc740 \uacc4\uc0b0 \uaddc\uce59\uacfc \uc815\ub9ac\ub97c \uc0ac\uc6a9\ud560 \uc218 \uc788\uac8c \ud55c \uac83\uc740 \ubd84\uba85\ud558\ub2e4.<\/p>\n<p>\ud655\ub960\uc758 \uc758\ubbf8\ub97c \ub458\ub7ec\uc2fc \ub17c\uc7c1\uc5d0\ub294 \uc5ec\ub7ec \uc785\uc7a5\uc774 \uc788\uc5c8\ub2e4. \uc874 \ubca4(John Venn)\uc740 1866\ub144 \u300e\uc6b0\uc5f0\uc758 \ub17c\ub9ac\u300f(<i>The Logic of Chance<\/i>)\uc5d0\uc11c \ud655\ub960\uc744 \uac00\uc0c1\uc801\uc778 \ubb34\ud55c \uacc4\uc5f4\uc5d0\uc11c \ub098\ud0c0\ub098\ub294 \uc0c1\ub300\ube48\ub3c4\uc640 \uc5f0\uacb0\ud558\ub294 \uacac\ud574\ub97c \uccb4\uacc4\uc801\uc73c\ub85c \uc804\uac1c\ud588\ub2e4.<a class=\"math-abs-series-ref\" title=\"Venn, J. (1866). The Logic of Chance: An Essay on the Foundations and Province of the Theory of Probability. London: Macmillan. Google Books.\" href=\"#ref-Venn1866\">[Venn1866]<\/a> \ub9ac\ud558\ub974\ud2b8 \ud3f0 \ubbf8\uc81c\uc2a4(Richard von Mises)\ub294 1919\ub144 \uc774\ub7ec\ud55c \ube48\ub3c4\uc801 \uc0dd\uac01\uc744 <span class=\"defined\">\uc9d1\ud569\uccb4<\/span>(Kollektiv)\ub77c\ub294 \uac1c\ub150\uc73c\ub85c \uc815\uc2dd\ud654\ud558\ub824 \ud588\ub2e4. \ud3f0 \ubbf8\uc81c\uc2a4\uc758 \uc9d1\ud569\uccb4\ub294 \uacb0\uacfc\ub4e4\uc758 \ubb34\ud55c\uc218\uc5f4\ub85c\uc11c, \uac01 \uacb0\uacfc\uc758 \uc0c1\ub300\ube48\ub3c4\uac00 \uadf9\ud55c\uc744 \uac00\uc838\uc57c \ud560 \ubfd0 \uc544\ub2c8\ub77c \ubbf8\ub798 \uacb0\uacfc\ub97c \ubbf8\ub9ac \uc0ac\uc6a9\ud558\uc9c0 \uc54a\ub294 \ud5c8\uc6a9 \uac00\ub2a5\ud55c \uc120\ud0dd\uaddc\uce59\uc73c\ub85c \ubd80\ubd84\uc218\uc5f4\uc744 \uace8\ub77c\ub3c4 \uac19\uc740 \uadf9\ud55c\uc774 \uc720\uc9c0\ub418\uc5b4\uc57c \ud588\ub2e4. \uc5b4\ub5a4 \uc120\ud0dd\uaddc\uce59\uc744 \ud5c8\uc6a9\ud560 \uac83\uc778\uc9c0\uc640 \uc774 \uc870\uac74\uc774 \ucda9\ubd84\ud55c \ubb34\uc791\uc704\uc131\uc744 \ud45c\ud604\ud558\ub294\uc9c0\ub294 \uc774\ud6c4 \ub17c\uc7c1\uc758 \ub300\uc0c1\uc774 \ub418\uc5c8\ub2e4.<a class=\"math-abs-series-ref\" title=\"von Mises, R. (1919). \u201cGrundlagen der Wahrscheinlichkeitsrechnung.\u201d Mathematische Zeitschrift, 5, 52\u201399.\" href=\"#ref-VonMises1919\">[VonMises1919]<\/a><\/p>\n<p>\uc8fc\uad00\uc801 \ud574\uc11d\ub3c4 \ubc1c\uc804\ud588\ub2e4. \ube0c\ub8e8\ub178 \ub370 \ud53c\ub124\ud2f0(Bruno de Finetti)\ub294 \ud655\ub960\uc744 \ubd88\ud655\uc2e4\ud55c \uba85\uc81c\uc5d0 \ub300\ud55c \uac1c\uc778\uc758 \ubbff\uc74c\uc758 \uc815\ub3c4\ub85c \ud574\uc11d\ud558\uace0, \uadf8 \ubbff\uc74c\uc744 \uc77c\uad00\ub41c \ubca0\ud305 \uac00\uaca9\uacfc \uc5f0\uacb0\ud588\ub2e4.<a class=\"math-abs-series-ref\" title=\"de Finetti, B. (1931). \u201cSul significato soggettivo della probabilit\u00e0.\u201d Fundamenta Mathematicae, 17, 298\u2013329.\" href=\"#ref-DeFinetti1931\">[DeFinetti1931]<\/a> <!-- \ub2e4\ub9cc \ud655\ub960 \ud574\uc11d\uc758 \uc5ed\uc0ac\ub97c \ube48\ub3c4\uc8fc\uc758\uc640 \uc8fc\uad00\uc8fc\uc758 \ub450 \uc9c4\uc601\uc758 \uc815\uba74 \ub300\uacb0\ub85c \ub2e8\uc21c\ud654\ud560 \uc218\ub294 \uc5c6\ub2e4. \ub2f9\uc2dc\uc5d0\ub3c4 \uace0\uc804\uc801 \ud574\uc11d\uacfc \ub17c\ub9ac\uc801 \ud574\uc11d \ub4f1\uc774 \uacbd\uc7c1\ud588\uace0, \ub4a4\uc5d0\ub294 \uc131\ud5a5 \ud574\uc11d\uacfc \ucd5c\uc120\uccb4\uacc4 \ud574\uc11d \uac19\uc740 \uacac\ud574\ub3c4 \ubc1c\uc804\ud588\ub2e4. \ud655\ub960 \ud574\uc11d\uc5d0 \uad00\ud55c \ub17c\uc7c1\uc740 \uc624\ub298\ub0a0\uc5d0\ub3c4 \uacc4\uc18d\ub41c\ub2e4.<a class=\"math-abs-series-ref\" title=\"H\u00e1jek, A. (2023). \u201cInterpretations of Probability.\u201d In E. N. Zalta &amp; U. Nodelman (Eds.), The Stanford Encyclopedia of Philosophy (Winter 2023 ed.). Metaphysics Research Lab, Stanford University.\" href=\"#ref-Hajek2023\">[Hajek2023]<\/a> --><\/p>\n<p>[\ud790\ubca0\ub974\ud2b8\uac00 1900\ub144 \ud30c\ub9ac \uc138\uacc4\uc218\ud559\uc790\ub300\ud68c \uac15\uc5f0\uacfc \ub4a4\uc774\uc740 \ucd9c\ud310\uc744 \ud1b5\ud574 \uc81c\uc2dc\ud55c \ubb38\uc81c \ubaa9\ub85d\uc758 \uc5ec\uc12f \ubc88\uc9f8 \ubb38\uc81c\ub294 \u201c\ubb3c\ub9ac\ud559 \uacf5\ub9ac\uc758 \uc218\ud559\uc801 \ucde8\uae09\u201d\uc774\uc5c8\ub2e4. \ud790\ubca0\ub974\ud2b8\ub294 \ud2b9\ud788 \ud655\ub960\ub860\uacfc \uc5ed\ud559\uc744 \uacf5\ub9ac\uc801\uc73c\ub85c \ub2e4\ub8f0 \uac83\uc744 \uc694\uccad\ud588\ub2e4. \ucf5c\ubaa8\uace0\ub85c\ud504\uc758 \uacf5\ub9ac\ud654\ub294 \uc774 \uac00\uc6b4\ub370 \ud655\ub960\ub860 \ubd80\ubd84\uc5d0 \ub300\ud55c \ub9e4\uc6b0 \uc911\uc694\ud55c \uc751\ub2f5\uc73c\ub85c \ud3c9\uac00\ud560 \uc218 \uc788\ub2e4. <!-- \uadf8\ub7ec\ub098 \ud790\ubca0\ub974\ud2b8\uc758 \uc5ec\uc12f \ubc88\uc9f8 \ubb38\uc81c\ub294 \uc5ed\ud559\uacfc \uc5f0\uc18d\uccb4 \uadf9\ud55c\uae4c\uc9c0 \ud3ec\ud568\ud558\ub294 \ub354 \ub113\uc740 \uc5f0\uad6c\uacc4\ud68d\uc774\ubbc0\ub85c, \ucf5c\ubaa8\uace0\ub85c\ud504\uac00 \ubb38\uc81c \uc804\uccb4\ub97c \ud574\uacb0\ud588\ub2e4\uace0 \ub9d0\ud574\uc11c\ub294 \uc548 \ub41c\ub2e4.<a class=\"math-abs-series-ref\" title=\"Hilbert, D. (1902). \u201cMathematical Problems.\u201d Bulletin of the American Mathematical Society, 8, 437\u2013479.\" href=\"#ref-Hilbert1902\">[Hilbert1902]<\/a><a class=\"math-abs-series-ref\" title=\"Gorban, A. N. (2018). \u201cHilbert\u2019s Sixth Problem: The Endless Road to Rigour.\u201d Philosophical Transactions of the Royal Society A, 376(2118), 20170238.\" href=\"#ref-Gorban2018\">[Gorban2018]<\/a>] --><\/p>\n<p>\uc548\ub4dc\ub808\uc774 \ucf5c\ubaa8\uace0\ub85c\ud504(Andrey Kolmogorov)\ub294 1933\ub144 \u300e\ud655\ub960\ub860\uc758 \uae30\ucd08\uac1c\ub150\u300f(<i>Grundbegriffe der Wahrscheinlichkeitsrechnung<\/i>)\uc5d0\uc11c \ud655\ub960\ub860\uc758 \uce21\ub3c4\ub860\uc801 \ud615\uc2dd\uc744 \uccb4\uacc4\uc801\uc73c\ub85c \uc81c\uc2dc\ud588\ub2e4.<a class=\"math-abs-series-ref\" title=\"Kolmogorov, A. N. (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung. Berlin: Julius Springer.\" href=\"#ref-Kolmogorov1933\">[Kolmogorov1933]<\/a> \uc774\ub294 \uc544\ubb34 \uc120\ud589 \uc791\uc5c5 \uc5c6\uc774 \uc0c8 \uc774\ub860\uc744 \ub9cc\ub4e0 \uc77c\uc774 \uc544\ub2c8\ub77c, \ubcf4\ub810\u00b7\ub974\ubca0\uadf8\u00b7\ub2e4\ub2c8\uc5d8\u00b7\ud504\ub808\uc170\ub97c \ube44\ub86f\ud55c \uc5f0\uad6c\uc790\ub4e4\uc774 \ubc1c\uc804\uc2dc\ud0a8 \uce21\ub3c4\uc640 \uc801\ubd84\uc758 \uc544\uc774\ub514\uc5b4\ub97c \ud558\ub098\uc758 \ud655\ub960\ub860 \uccb4\uacc4\ub85c \uc885\ud569\ud55c \uc791\uc5c5\uc774\uc5c8\ub2e4.<a class=\"math-abs-series-ref\" title=\"Shafer, G., &amp; Vovk, V. (2006). \u201cThe Sources of Kolmogorov\u2019s Grundbegriffe.\u201d Statistical Science, 21(1), 70\u201398.\" href=\"#ref-ShaferVovk2006\">[ShaferVovk2006]<\/a><\/p>\n<div class=\"box theorem\">\n<p><span class=\"theorem\">\uc815\uc758: \ud655\ub960\uacf5\uac04<\/span><br \/>\n\uc9d1\ud569 \\(\\Omega\\), \uadf8 \uc704\uc758 \uc2dc\uadf8\ub9c8\ub300\uc218 \\(\\mathcal F\\), \ud568\uc218 \\(P:\\mathcal F\\to[0,1]\\)\uac00 \ub2e4\uc74c \uc870\uac74\uc744 \ub9cc\uc871\ud55c\ub2e4\uace0 \ud558\uc790.<\/p>\n<ul>\n<li>\ubaa8\ub4e0 \\(A\\in\\mathcal F\\)\uc5d0 \ub300\ud558\uc5ec \\(P(A)\\ge0\\)\uc774\ub2e4.<\/li>\n<li>\\(P(\\Omega)=1\\)\uc774\ub2e4.<\/li>\n<li>\uc11c\ub85c\uc18c\uc778 \uc0ac\uac74\uc5f4 \\(A_1,A_2,\\ldots\\in\\mathcal F\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nP\\left(\\bigcup_{n=1}^{\\infty}A_n\\right)<br \/>\n=\\sum_{n=1}^{\\infty}P(A_n)<br \/>\n\\]<br \/>\n\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\uc774\ub54c \\(P\\)\ub97c <span class=\"defined\">\ud655\ub960\uce21\ub3c4<\/span>(probability measure), \uc21c\uc11c\uc0bc\uc911\ud56d \\((\\Omega,\\mathcal F,P)\\)\ub97c <span class=\"defined\">\ud655\ub960\uacf5\uac04<\/span>(probability space)\uc774\ub77c\uace0 \ud55c\ub2e4. \\(\\Omega\\)\uc758 \uc6d0\uc18c \\(\\omega\\)\ub294 <span class=\"defined\">\ud45c\ubcf8\uc810<\/span>(sample point) \ub610\ub294 <span class=\"defined\">\uacb0\uacfc<\/span>(outcome)\uc774\uace0, \\(\\mathcal F\\)\uc758 \uc6d0\uc18c\ub294 <span class=\"defined\">\uc0ac\uac74<\/span>(event)\uc774\ub2e4.<a class=\"math-abs-series-ref\" title=\"Kallenberg, O. (2021). Foundations of Modern Probability (3rd ed.). Springer.\" href=\"#ref-Kallenberg2021\">[Kallenberg2021]<\/a><\/p>\n<\/div>\n<p>\ucf5c\ubaa8\uace0\ub85c\ud504\uc758 \uc6d0\ub798 \uc81c\uc2dc\ub294 \uc624\ub298\ub0a0 \uad50\uacfc\uc11c\uc758 \ubb38\uc7a5\uacfc \uadf8\ub300\ub85c \uac19\uc9c0\ub294 \uc54a\uc558\ub2e4. \uadf8\ub294 \uc2dc\uadf8\ub9c8\ub300\uc218 \ub300\uc2e0 \uc9d1\ud569\uccb4(field of sets)\ub97c \uc815\uc758\uc5ed\uc73c\ub85c \uc0bc\uace0, \ube44\uc74c\uc131\u00b7\uc815\uaddc\ud654\u00b7\uc720\ud55c\uac00\ubc95\uc131\uc744 \uc81c\uc2dc\ud55c \ub4a4 \uac10\uc18c\ud558\ub294 \uc0ac\uac74\uc5f4\uc5d0 \uad00\ud55c \uc5f0\uc18d\uc131 \uacf5\ub9ac\ub97c \ucd94\uac00\ud588\ub2e4. \uc720\ud55c\uac00\ubc95\uc131\uacfc \uc774 \uc5f0\uc18d\uc131 \uacf5\ub9ac\ub97c \ud568\uaed8 \uc4f0\uba74, \uc11c\ub85c\uc18c\uc778 \uc9d1\ud569\uc5f4\uc758 \ud569\uc9d1\ud569\uc774 \uadf8 \uc9d1\ud569\uccb4\uc5d0 \uc18d\ud558\ub294 \uacbd\uc6b0 \uac00\uc0b0\uac00\ubc95\uc131\uc744 \uc5bb\ub294\ub2e4. \uc624\ub298\ub0a0\uc758 \uc815\uc758\ub294 \uc815\uc758\uc5ed\uc744 \uc2dc\uadf8\ub9c8\ub300\uc218\ub85c \ub450\uace0 \uac00\uc0b0\uac00\ubc95\uc131\uc744 \uc9c1\uc811 \uc694\uad6c\ud558\ub294 \ud45c\uc900 \ud615\uc2dd\uc774\ub2e4.<a class=\"math-abs-series-ref\" title=\"Kolmogorov, A. N. (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung. Berlin: Julius Springer.\" href=\"#ref-Kolmogorov1933\">[Kolmogorov1933]<\/a><a class=\"math-abs-series-ref\" title=\"Shafer, G., &amp; Vovk, V. (2006). \u201cThe Sources of Kolmogorov\u2019s Grundbegriffe.\u201d Statistical Science, 21(1), 70\u201398.\" href=\"#ref-ShaferVovk2006\">[ShaferVovk2006]<\/a><\/p>\n<p>\uacf5\ub9ac\uacc4 \uc790\uccb4\ub294 \ud655\ub960\uc774 \uc7a5\uae30 \uc0c1\ub300\ube48\ub3c4\uc778\uc9c0, \ubb3c\ub9ac\uc801 \uc131\ud5a5\uc778\uc9c0, \ud569\ub9ac\uc801 \ubbff\uc74c\uc758 \uc815\ub3c4\uc778\uc9c0\ub97c \uacb0\uc815\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc77c\ub2e8 \uc5b4\ub5a4 \ubaa8\ud615\uc774 \uc704 \uc870\uac74\uc744 \ub9cc\uc871\ud558\uba74 \uce21\ub3c4\uc758 \uc5f0\uc18d\uc131\uacfc \uc218\ub834\uc815\ub9ac \uac19\uc740 \uacb0\uacfc\ub97c \uc801\uc6a9\ud560 \uc218 \uc788\ub2e4. <!-- \uadf8\ub7ec\ub098 \uc774\uac83\uc774 \ud655\ub960\uc758 \ud574\uc11d\uc774\ub098 \uacbd\ud5d8\uc801 \uc801\uc6a9 \ubb38\uc81c\ub97c \uc5c6\uc560 \uc8fc\ub294 \uac83\uc740 \uc544\ub2c8\ub2e4. --> \ucf5c\ubaa8\uace0\ub85c\ud504\uc758 \uc6d0\uc804\uc5d0\ub3c4 \uc218\ud559\uc801 \uacf5\ub9ac\uacc4\uc640 \uacbd\ud5d8 \uc138\uacc4\ub97c \uc5f0\uacb0\ud558\ub294 \ud574\uc11d\uc774 \ubcc4\ub3c4\ub85c \ub4e4\uc5b4 \uc788\ub2e4. <!-- \ub610\ud55c \ub370 \ud53c\ub124\ud2f0 \uc790\uc2e0\uc740 \uac00\uc0b0\uac00\ubc95\uc131\uc744 \ud544\uc218\uc801\uc778 \ud569\ub9ac\uc131 \uc870\uac74\uc73c\ub85c \ubc1b\uc544\ub4e4\uc774\uc9c0 \uc54a\uace0 \uc720\ud55c\uac00\ubc95\uc801 \ud655\ub960\uc744 \uc120\ud638\ud588\uc73c\ubbc0\ub85c, \ubaa8\ub4e0 \uc8fc\uad00\uc801 \ud655\ub960 \ud574\uc11d\uc774 \uadf8\ub300\ub85c \ucf5c\ubaa8\uace0\ub85c\ud504 \uacf5\ub9ac\uacc4\ub97c \ucc44\ud0dd\ud55c\ub2e4\uace0 \ub9d0\ud560 \uc218\ub3c4 \uc5c6\ub2e4.<a class=\"math-abs-series-ref\" title=\"Kolmogorov, A. N. (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung. Berlin: Julius Springer.\" href=\"#ref-Kolmogorov1933\">[Kolmogorov1933]<\/a><a class=\"math-abs-series-ref\" title=\"Howson, C. (2008). \u201cDe Finetti, Countable Additivity, Consistency and Coherence.\u201d The British Journal for the Philosophy of Science, 59(1), 1\u201323.\" href=\"#ref-Howson2008\">[Howson2008]<\/a> --><\/p>\n<h4>\ud655\ub960\ub860\uc758 \uc0ac\uc804<\/h4>\n<p>\uc9d1\ud569\uc758 \uce21\ub3c4\uc640 \uc0ac\uac74\uc758 \ud655\ub960, \ud568\uc218\uc758 \uc801\ubd84\uacfc \ud655\ub960\ubcc0\uc218\uc758 \uae30\ub313\uac12 \uc0ac\uc774\uc758 \uc720\uc0ac\uc131\uc740 \ucf5c\ubaa8\uace0\ub85c\ud504 \uc774\uc804\uc5d0\ub3c4 \uc54c\ub824\uc838 \uc788\uc5c8\ub2e4. 1933\ub144\uc758 \uc800\uc11c\ub294 \uc774\ub7ec\ud55c \ub300\uc751\uc744 \uc77c\uad00\ub41c \ud655\ub960\ub860 \uccb4\uacc4 \uc548\uc5d0 \uc815\ucc29\uc2dc\ud0a4\ub294 \ub370 \uacb0\uc815\uc801\uc778 \uc5ed\ud560\uc744 \ud588\ub2e4. <!-- \ub2e4\uc74c \ub0b4\uc6a9\uc740 \ub450 \ubd84\uc57c \uc0ac\uc774\uc758 \u201c\uc644\ubcbd\ud55c \uc77c\ub300\uc77c \ub300\uc751\u201d\uc774\ub77c\uae30\ubcf4\ub2e4 \uba87 \uac00\uc9c0 \ud575\uc2ec \ub300\uc751\uc73c\ub85c \uc774\ud574\ud574\uc57c \ud55c\ub2e4.<a class=\"math-abs-series-ref\" title=\"Kolmogorov, A. N. (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung. Berlin: Julius Springer.\" href=\"#ref-Kolmogorov1933\">[Kolmogorov1933]<\/a><a class=\"math-abs-series-ref\" title=\"Shafer, G., &amp; Vovk, V. (2006). \u201cThe Sources of Kolmogorov\u2019s Grundbegriffe.\u201d Statistical Science, 21(1), 70\u201398.\" href=\"#ref-ShaferVovk2006\">[ShaferVovk2006]<\/a> --><\/p>\n<p>\uac00\uce21\uacf5\uac04 \\((S,\\Sigma)\\)\uc5d0\uc11c \uc2e4\uc218\ub85c \uac00\ub294 \ud568\uc218 \\(f:S\\to\\mathbb R\\)\ub97c \uc0dd\uac01\ud558\uc790. \uc2e4\uc218\uc758 \ubcf4\ub810 \uc2dc\uadf8\ub9c8\ub300\uc218\ub97c \\(\\mathcal B(\\mathbb R)\\)\ub77c \ud560 \ub54c, \ubaa8\ub4e0 \\(B\\in\\mathcal B(\\mathbb R)\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nf^{-1}(B)\\in\\Sigma<br \/>\n\\]<br \/>\n\uc774\uba74 \\(f\\)\ub97c <span class=\"defined\">\uac00\uce21\ud568\uc218<\/span>(measurable function)\ub77c\uace0 \ud55c\ub2e4. \ubcf4\ub810 \uc2dc\uadf8\ub9c8\ub300\uc218\ub294 \ubc18\uc9c1\uc120\ub4e4\ub85c \uc0dd\uc131\ub418\ubbc0\ub85c, \ubaa8\ub4e0 \\(t\\in\\mathbb R\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nf^{-1}((-\\infty,t])\\in\\Sigma<br \/>\n\\]<br \/>\n\uc778\uc9c0 \ud655\uc778\ud558\ub294 \uac83\uacfc \ub3d9\uce58\uc774\ub2e4. [\uac00\uce21\uc131\uc758 \uc6d0\uc0c1 \uc870\uac74\uc740 \u201c\uc5f4\ub9b0\uc9d1\ud569\uc758 \uc6d0\uc0c1\uc774 \uc5f4\ub9b0\uc9d1\ud569\u201d\uc774\ub77c\ub294 \uc5f0\uc18d\uc131\uc758 \uc815\uc758\uc640 \ud615\uc2dd\uc801\uc73c\ub85c \ub2ee\uc558\ub2e4. \uadf8\ub7ec\ub098 \ub450 \uc870\uac74\uc740 \ub3d9\uc77c\ud558\uc9c0 \uc54a\ub2e4. \uc5f0\uc18d\uc131\uc740 \uc704\uc0c1\uc758 \uc5f4\ub9b0\uc9d1\ud569\uc5d0 \uad00\ud55c \uc870\uac74\uc774\uace0, \uac00\uce21\uc131\uc740 \uc2dc\uadf8\ub9c8\ub300\uc218\uc5d0 \uc18d\ud558\ub294 \uc9d1\ud569\uc5d0 \uad00\ud55c \uc870\uac74\uc774\ub2e4. \uc704\uc0c1\uacf5\uac04\uc5d0 \ubcf4\ub810 \uc2dc\uadf8\ub9c8\ub300\uc218\ub97c \uc8fc\uba74 \uc5f0\uc18d\ud568\uc218\ub294 \uac00\uce21\uc774\uc9c0\ub9cc, \uac00\uce21\ud568\uc218\uac00 \ubc18\ub4dc\uc2dc \uc5f0\uc18d\uc778 \uac83\uc740 \uc544\ub2c8\ub2e4.]<\/p>\n<p>\ud655\ub960\uacf5\uac04 \\((\\Omega,\\mathcal F,P)\\)\uc5d0\uc11c \\((\\mathbb R,\\mathcal B(\\mathbb R))\\)\ub85c \uac00\ub294 \uac00\uce21\ud568\uc218 \\(X:\\Omega\\to\\mathbb R\\)\ub97c <span class=\"defined\">\ud655\ub960\ubcc0\uc218<\/span>(random variable)\ub77c\uace0 \ud55c\ub2e4.<a class=\"math-abs-series-ref\" title=\"Kallenberg, O. (2021). Foundations of Modern Probability (3rd ed.). Springer.\" href=\"#ref-Kallenberg2021\">[Kallenberg2021]<\/a><\/p>\n<p>\uac00\uce21\uc131\ub9cc\uc73c\ub85c \ubaa8\ub4e0 \uc2e4\uc218\uac12 \ud655\ub960\ubcc0\uc218\uac00 \uc720\ud55c\ud55c \ub974\ubca0\uadf8 \uc801\ubd84\uc744 \uac16\ub294 \uac83\uc740 \uc544\ub2c8\ub2e4. \\(X\\ge0\\)\uc774\uba74<br \/>\n\\[<br \/>\nE[X]=\\int_\\Omega X\\,dP<br \/>\n\\]<br \/>\n\ub97c \\(+\\infty\\)\uae4c\uc9c0 \ud5c8\uc6a9\ud558\ub294 \ud655\uc7a5\ub41c \uac12\uc73c\ub85c \uc815\uc758\ud560 \uc218 \uc788\ub2e4. \ubd80\ud638\uac00 \ubc14\ub00c\ub294 \ud655\ub960\ubcc0\uc218\uac00 \uc720\ud55c\ud55c \uae30\ub313\uac12\uc744 \uac00\uc9c0\ub824\uba74<br \/>\n\\[<br \/>\nE[|X|]=\\int_\\Omega |X|\\,dP&lt;\\infty<br \/>\n\\]<br \/>\n\uc774\uc5b4\uc57c \ud55c\ub2e4. \uc774 \uc870\uac74\uc744 \ub9cc\uc871\ud558\ub294 \\(X\\)\uc758 <span class=\"defined\">\uae30\ub313\uac12<\/span>(expectation)\uc740<br \/>\n\\[<br \/>\nE[X]=\\int_\\Omega X\\,dP<br \/>\n\\]<br \/>\n\ub85c \uc815\uc758\ub41c\ub2e4.<a class=\"math-abs-series-ref\" title=\"Kallenberg, O. (2021). Foundations of Modern Probability (3rd ed.). Springer.\" href=\"#ref-Kallenberg2021\">[Kallenberg2021]<\/a><\/p>\n<p><!--\n\n\n<p>[\ube44\uc74c\uc218 \uac00\uce21\ud568\uc218\uc758 \ub974\ubca0\uadf8 \uc801\ubd84\uc740 \uadf8 \ud568\uc218 \uc774\ud558\uc778 \ube44\uc74c\uc218 \ub2e8\uc21c\ud568\uc218\ub4e4\uc758 \uc801\ubd84\uac12\uc758 \uc0c1\ud55c\uc73c\ub85c \uc815\uc758\ud560 \uc218 \uc788\ub2e4. \uc77c\ubc18 \uc2e4\uc218\uac12 \uac00\uce21\ud568\uc218\ub294 \uc591\uc758 \ubd80\ubd84\uacfc \uc74c\uc758 \ubd80\ubd84\uc73c\ub85c \ub098\ub204\uc5b4 \uc801\ubd84\uc744 \uc815\uc758\ud55c\ub2e4. \u201c\ub9ac\ub9cc \uc801\ubd84\uc740 \\(x\\)\ucd95\uc744 \ub098\ub204\uace0 \ub974\ubca0\uadf8 \uc801\ubd84\uc740 \\(y\\)\ucd95\uc744 \ub098\ub208\ub2e4\u201d\ub294 \uc124\uba85\uc740 \ud568\uc218\uac12\uc758 \ubc94\uc704\uc5d0 \ub300\uc751\ud558\ub294 \uc6d0\uc0c1 \uc9d1\ud569\uc758 \uce21\ub3c4\ub97c \uc774\uc6a9\ud55c\ub2e4\ub294 \uc810\uc744 \ub098\ud0c0\ub0b4\ub294 \uc720\uc6a9\ud55c \uc9c1\uad00\uc774\ub2e4. \uadf8\ub7ec\ub098 \uc774\uac83\uc774 \ub974\ubca0\uadf8 \uc801\ubd84\uc758 \uc5c4\ubc00\ud55c \uc815\uc758 \uc790\uccb4\ub294 \uc544\ub2c8\ub2e4.]<\/p>\n\n\n--><\/p>\n<p>\uc608\ub97c \ub4e4\uc5b4 \\([0,1]\\)\uc5d0 \ub974\ubca0\uadf8 \uce21\ub3c4\ub97c \ud655\ub960\uce21\ub3c4\ub85c \ub450\uba74 \ub514\ub9ac\ud074\ub808 \ud568\uc218\ub294 \uc720\ub9ac\uc218 \uc9d1\ud569\uc758 \uc9c0\uc2dc\ud568\uc218\uc774\ub2e4. \uc720\ub9ac\uc218 \uc9d1\ud569\uc758 \uce21\ub3c4\uac00 \\(0\\)\uc774\ubbc0\ub85c<br \/>\n\\[<br \/>\n\\int_{[0,1]}D\\,dm=0<br \/>\n\\]<br \/>\n\uc774\ub2e4. \ub9ac\ub9cc \uc801\ubd84\uc740 \uc874\uc7ac\ud558\uc9c0 \uc54a\uc9c0\ub9cc \ub974\ubca0\uadf8 \uc801\ubd84\uc740 \uc874\uc7ac\ud558\ub294 \uad6c\uccb4\uc801\uc778 \uc608\uc774\ub2e4. \uc720\ud55c \ud45c\ubcf8\uacf5\uac04\uc5d0\uc11c \\(\\mathcal F=\\mathcal P(\\Omega)\\)\uc778 \uacbd\uc6b0\uc5d0\ub294 \ub974\ubca0\uadf8 \uc801\ubd84\uc5d0 \uc758\ud55c \uae30\ub313\uac12\uc774 \uc775\uc219\ud55c \uac00\uc911\ud569<br \/>\n\\[<br \/>\nE[X]=\\sum_{\\omega\\in\\Omega}X(\\omega)P(\\{\\omega\\})<br \/>\n\\]<br \/>\n\uc73c\ub85c \ud658\uc6d0\ub41c\ub2e4.<\/p>\n<p>\ub450 \uc0ac\uac74 \\(A,B\\in\\mathcal F\\)\uac00<br \/>\n\\[<br \/>\nP(A\\cap B)=P(A)P(B)<br \/>\n\\]<br \/>\n\ub97c \ub9cc\uc871\ud558\uba74 \ub450 \uc0ac\uac74\uc774 <span class=\"defined\">\ub3c5\ub9bd<\/span>(independent)\uc774\ub77c\uace0 \ud55c\ub2e4. \ub450 \uc2e4\uc218\uac12 \ud655\ub960\ubcc0\uc218 \\(X,Y\\)\uc758 \uc8fc\ubcc0\ubd84\ud3ec\uc640 \uacb0\ud569\ubd84\ud3ec\ub97c \uac01\uac01<br \/>\n\\[<br \/>\n\\mu_X=P\\circ X^{-1},\\qquad<br \/>\n\\mu_Y=P\\circ Y^{-1},\\qquad<br \/>\n\\mu_{(X,Y)}=P\\circ(X,Y)^{-1}<br \/>\n\\]<br \/>\n\ub85c \ub450\uc790. \\(X\\)\uc640 \\(Y\\)\uac00 \ub3c5\ub9bd\uc774\ub77c\ub294 \uac83\uc740 \ubaa8\ub4e0 \ubcf4\ub810\uc9d1\ud569 \\(A,B\\subseteq\\mathbb R\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nP(X\\in A,\\,Y\\in B)=P(X\\in A)P(Y\\in B)<br \/>\n\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4\ub294 \ub73b\uc774\uba70, \uc774\ub294<br \/>\n\\[<br \/>\n\\mu_{(X,Y)}=\\mu_X\\otimes\\mu_Y<br \/>\n\\]<br \/>\n\uc640 \ub3d9\uce58\uc774\ub2e4.<a class=\"math-abs-series-ref\" title=\"Kallenberg, O. (2021). Foundations of Modern Probability (3rd ed.). Springer.\" href=\"#ref-Kallenberg2021\">[Kallenberg2021]<\/a><\/p>\n<p>[\ub450 \ud655\ub960\uce21\ub3c4 \\(\\mu_1,\\mu_2\\)\uac00 \uac01\uac01 \\((S_1,\\Sigma_1)\\), \\((S_2,\\Sigma_2)\\) \uc704\uc5d0 \uc8fc\uc5b4\uc9c0\uba74, \uacf1 \uc2dc\uadf8\ub9c8\ub300\uc218 \\(\\Sigma_1\\otimes\\Sigma_2\\) \uc704\uc5d0\ub294<br \/>\n\\[<br \/>\n(\\mu_1\\otimes\\mu_2)(A_1\\times A_2)<br \/>\n=\\mu_1(A_1)\\mu_2(A_2)<br \/>\n\\]<br \/>\n\ub97c \ub9cc\uc871\ud558\ub294 \uc720\uc77c\ud55c <span class=\"defined\">\uacf1\uce21\ub3c4<\/span>(product measure)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc774 \uad6c\uc131\uc5d0 \ub450 \uce21\ub3c4\uacf5\uac04\uc774 \ubbf8\ub9ac \u201c\ub3c5\ub9bd\uc801\u201d\uc774\ub77c\ub294 \uc804\uc81c\ub294 \ud544\uc694\ud558\uc9c0 \uc54a\ub2e4. \ub3c5\ub9bd\uc131\uc740 \uacf5\ub3d9 \ud655\ub960\uacf5\uac04 \uc704\uc758 \ud655\ub960\ubcc0\uc218\ub098 \uc2dc\uadf8\ub9c8\ub300\uc218\uc5d0 \ub300\ud574 \uc815\uc758\ub418\ub294 \ubcc4\ub3c4\uc758 \uc131\uc9c8\uc774\ub2e4.<a class=\"math-abs-series-ref\" title=\"Kallenberg, O. (2021). Foundations of Modern Probability (3rd ed.). Springer.\" href=\"#ref-Kallenberg2021\">[Kallenberg2021]<\/a>]<\/p>\n<p>\uc774 \ub300\uc751\uc740 \ud655\ub960\ub860\uc5d0 \uce21\ub3c4\ub860\uc801 \uae30\ucd08\ub97c \uc81c\uacf5\ud55c\ub2e4. <!-- \uadf8\ub807\ub2e4\uace0 \ud655\ub960\ub860 \uc804\uccb4\uac00 \uc77c\ubc18 \uce21\ub3c4\ub860\uc5d0\uc11c \uc6a9\uc5b4\ub9cc \ubc14\uafbc \ubd84\uc57c\ub77c\ub294 \ub73b\uc740 \uc544\ub2c8\ub2e4. --> \uc870\uac74\ubd80\ud655\ub960, \ub3c5\ub9bd\uc131, \ud655\ub960\uacfc\uc815, \uacbd\ud5d8\uc801 \ubaa8\ud615\ud654\ub294 \ud655\ub960\ub860\uc5d0 \uace0\uc720\ud55c \uad6c\uc870\uc640 \ubb38\uc81c\ub97c \ud615\uc131\ud55c\ub2e4.<a class=\"math-abs-series-ref\" title=\"Kolmogorov, A. N. (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung. Berlin: Julius Springer.\" href=\"#ref-Kolmogorov1933\">[Kolmogorov1933]<\/a><a class=\"math-abs-series-ref\" title=\"Kallenberg, O. (2021). Foundations of Modern Probability (3rd ed.). Springer.\" href=\"#ref-Kallenberg2021\">[Kallenberg2021]<\/a><\/p>\n<h4>\uc6b0\uc5f0\uc758 \ubcf8\uc131\uc744 \ud655\uc815\ud558\uc9c0 \uc54a\uace0 \ud615\uc2dd\uc744 \uc138\uc6b0\ub2e4<\/h4>\n<p>\uc9c0\uae08\uae4c\uc9c0 \uae30\ud558\ud559\uc801 \uae38\uc774\ub97c \ubcf5\uc7a1\ud55c \uc9d1\ud569\uc73c\ub85c \ud655\uc7a5\ud558\ub294 \uacfc\uc815\uc744 \ucd94\uc801\ud588\ub2e4. \ub514\ub9ac\ud074\ub808 \ud568\uc218\ub294 \ud604\ub300\uc801 \uad00\uc810\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84\uc758 \uc81c\ud55c\uc744 \ubcf4\uc5ec \uc900\ub2e4. \ub974\ubca0\uadf8\ub294 \uc870\ub974\ub2f9\uacfc \ubcf4\ub810 \ub4f1\uc758 \uc120\ud589 \uc5f0\uad6c\ub97c \ubc14\ud0d5\uc73c\ub85c 1902\ub144\uc5d0 \uce21\ub3c4\uc640 \uc801\ubd84\uc744 \ud655\uc7a5\ud588\uace0, \ube44\ud0c8\ub9ac\ub294 1905\ub144\uc5d0 \uc120\ud0dd\uacf5\ub9ac\ub97c \uc774\uc6a9\ud55c \uc9d1\ud569\uc73c\ub85c \uadf8 \uae38\uc774\ub97c \ubaa8\ub4e0 \ubd80\ubd84\uc9d1\ud569\uc5d0\uae4c\uc9c0 \ud655\uc7a5\ud560 \uc218 \uc5c6\uc74c\uc744 \uc120\uba85\ud558\uac8c \ub4dc\ub7ec\ub0c8\ub2e4. \uce74\ub77c\ud14c\uc624\ub3c4\ub9ac\uc758 \ud310\uc815\ubc95\uacfc \ucd94\uc0c1 \uce21\ub3c4\uacf5\uac04\uc758 \uacf5\ub9ac\ub294 \uce21\uc815\ud560 \uc9d1\ud569\uc871\uacfc \uce21\ub3c4 \ud568\uc218\ub97c \ubd84\ub9ac\ud574 \ub2e4\ub8e8\ub294 \uc77c\ubc18\uc801\uc778 \uc5b8\uc5b4\ub97c \uc81c\uacf5\ud588\ub2e4.<\/p>\n<p>\ucf5c\ubaa8\uace0\ub85c\ud504\uc758 \uacf5\ub9ac\ud654\ub294 \uc11c\ub85c \ub2e4\ub978 \ud655\ub960 \ud574\uc11d\ub4e4\uc774 \uacf5\uc720\ud560 \uc218 \uc788\ub294 \uacf5\ud1b5\uc758 \uc218\ud559\uc801 \ud615\uc2dd\uc744 \ud655\ub9bd\ud588\ub2e4. \uc774 \ud615\uc2dd \uc548\uc5d0\uc11c \ud655\ub960\ubcc0\uc218\ub294 \uac00\uce21\ud568\uc218\ub85c, \uc801\ubd84 \uac00\ub2a5\ud55c \ud655\ub960\ubcc0\uc218\uc758 \uae30\ub313\uac12\uc740 \ub974\ubca0\uadf8 \uc801\ubd84\uc73c\ub85c, \ub3c5\ub9bd\uc131\uc740 \uacb0\ud569\ubd84\ud3ec\uc640 \uacf1\uce21\ub3c4\uc758 \uad00\uacc4\ub85c \ud45c\ud604\ub41c\ub2e4. \uc774 \uac1c\ub150\ub4e4\uc740 \ucf5c\ubaa8\uace0\ub85c\ud504 \uc774\uc804\uc5d0\ub3c4 \uc5f0\uad6c\ub418\uace0 \uc788\uc5c8\uc9c0\ub9cc, 1933\ub144\uc758 \uc800\uc11c\ub294 \uadf8\uac83\ub4e4\uc744 \ud604\ub300 \ud655\ub960\ub860\uc758 \uc77c\uad00\ub41c \uccb4\uacc4 \uc548\uc5d0 \uc815\ucc29\uc2dc\ud0a4\ub294 \ub370 \uacb0\uc815\uc801\uc778 \uc5ed\ud560\uc744 \ud588\ub2e4.<\/p>\n<p>\uc774 \uc5f0\uc7ac\uc5d0\uc11c\ub294 \uc774\ub7ec\ud55c \ucd94\uc0c1\ud654\uc758 \ud6a8\uacfc\ub97c <span class=\"defined\">\ub17c\uc7c1\uc758 \uc6b0\ud68c<\/span>\ub77c\uace0 \ubd80\ub974\uae30\ub85c \ud55c\ub2e4. \ub300\uc0c1\uc758 \uc874\uc7ac\ub860\uc801 \ubcf8\uc9c8\uc744 \ud558\ub098\ub85c \ud655\uc815\ud558\uc9c0 \uc54a\uc740 \ucc44, \uc5ec\ub7ec \ud574\uc11d\uc774 \uacf5\uc720\ud560 \uc218 \uc788\ub294 \ud615\uc2dd\uc801 \uc870\uac74\uc744 \ubd84\ub9ac\ud558\uc5ec \uc774\ub860\uc744 \uc804\uac1c\ud558\ub294 \ubc29\uc2dd\uc774\ub2e4. <!-- \ub2e4\ub9cc \uc774\ub97c \ucf5c\ubaa8\uace0\ub85c\ud504\uac00 \ubaa8\ub4e0 \ucca0\ud559\uc801 \ubb38\uc81c\ub97c \uacf5\ub9ac\uacc4 \ubc16\uc73c\ub85c \uc644\uc804\ud788 \ucd94\ubc29\ud588\ub2e4\ub294 \uc5ed\uc0ac\uc801 \uc8fc\uc7a5\uc73c\ub85c \ubc1b\uc544\ub4e4\uc5ec\uc11c\ub294 \uc548 \ub41c\ub2e4. --> \uadf8\uc758 \uc6d0\uc804\uc740 \uc218\ud559\uc801 \ud615\uc2dd\uacfc \uacbd\ud5d8 \uc138\uacc4\uc758 \uc5f0\uacb0\ub3c4 \ubcc4\ub3c4\ub85c \ub17c\uc758\ud588\ub2e4.<\/p>\n<p><!-- \ub610\ud55c \ud655\ub960\uc5d0\uc11c \ub0a8\ub294 \uc870\uac74\uc740 \ub2e8\uc21c\ud788 \\(P(\\Omega)=1\\) \ud558\ub098\uac00 \uc544\ub2c8\ub2e4. -->\ud655\ub960\ubaa8\ud615\uc758 \uc218\ud559\uc801 \ud575\uc2ec\uc740 \uc0ac\uac74\ub4e4\uc758 \uc2dc\uadf8\ub9c8\ub300\uc218\uc640 \uadf8 \uc704\uc5d0 \uc815\uc758\ub41c \ube44\uc74c\u00b7\uac00\uc0b0\uac00\ubc95\u00b7\uc815\uaddc\ud654 \uce21\ub3c4 \\(P\\)\uc774\ub2e4. \uacf5\ub9ac\uacc4\ub294 \uc774 \uad6c\uc870 \uc548\uc5d0\uc11c \ubb34\uc5c7\uc744 \uacc4\uc0b0\ud560 \uc218 \uc788\ub294\uc9c0\ub97c \uc54c\ub824 \uc8fc\uc9c0\ub9cc, \uc2e4\uc81c \ud604\uc0c1\uc5d0\uc11c \ubb34\uc5c7\uc744 \\(\\Omega\\), \\(\\mathcal F\\), \\(P\\)\ub85c \uc120\ud0dd\ud574\uc57c \ud558\ub294\uc9c0\uc640 \uadf8 \uc218\uce58\uac00 \ube48\ub3c4\u00b7\uc131\ud5a5\u00b7\ubbff\uc74c \uac00\uc6b4\ub370 \ubb34\uc5c7\uc744 \ub73b\ud558\ub294\uc9c0\ub294 \ubcc4\ub3c4\uc758 \ud574\uc11d \ubc0f \ubaa8\ud615\ud654 \ubb38\uc81c\ub85c \ub0a8\ub294\ub2e4.<\/p>\n<p>\uc9c0\uae08\uae4c\uc9c0 \uc138 \ud3b8\uc758 \uc5f0\uc7ac\uc5d0 \uac78\uccd0 \uc0b4\ud3b4\ubcf8 \ubca1\ud130\uacf5\uac04, \uc704\uc0c1\uacf5\uac04, \uce21\ub3c4\uacf5\uac04\uc740 \ud604\ub300 \uc218\ud559\uc758 \uc911\uc694\ud55c \uacf5\ub9ac\uc801 \uad6c\uc870\ub4e4\uc774\ub2e4. \uc624\ub298\ub0a0\uc758 \uad50\uacfc\uc11c\ub294 \ub300\uac1c \uc644\uc131\ub41c \uc815\uc758\uc640 \uc815\ub9ac\uc758 \uc5f0\uc5ed\uc801 \uc21c\uc11c\ub85c \uc774 \uad6c\uc870\ub4e4\uc744 \uc81c\uc2dc\ud558\uc9c0\ub9cc, \uc5ed\uc0ac\uc801 \ubc1c\uacac\uc740 \ud6e8\uc52c \ubcf5\uc7a1\ud55c \ubb38\uc81c\uc640 \uc2e4\ud328\ub97c \uac70\uccd0 \uc9c4\ud589\ub418\uc5c8\ub2e4. <a href=\"..\/math-abstraction-09-cognitive-process\/\">9\ubd80<\/a>\uc5d0\uc11c\ub294 \uc778\ub958\uac00 \uac1c\ub150\uc744 \ubc1c\uacac\ud55c \uc5ed\uc0ac\uc801 \uc21c\uc11c, \uc644\uc131\ub41c \uc218\ud559\uc744 \ubc30\uc5f4\ud558\ub294 \ub17c\ub9ac\uc801 \uc21c\uc11c, \uac1c\uc778\uc774 \uac1c\ub150\uc744 \uc775\ud788\ub294 \uc778\uc9c0\uc801 \uc21c\uc11c\uac00 \uc65c \uc11c\ub85c \uc5b4\uae0b\ub098\ub294\uc9c0 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n<h4>\ucc38\uace0\ubb38\ud5cc<\/h4>\n<ul>\n<li id=\"ref-Darboux1875\">[Darboux1875] Darboux, G. (1875). \u201cM\u00e9moire sur les fonctions discontinues.\u201d <i>Annales scientifiques de l\u2019\u00c9cole Normale Sup\u00e9rieure<\/i>, 2e s\u00e9rie, 4, 57\u2013112. <a href=\"https:\/\/doi.org\/10.24033\/asens.122\">https:\/\/doi.org\/10.24033\/asens.122<\/a>, <a href=\"https:\/\/www.numdam.org\/item\/ASENS_1875_2_4__57_0\/\">Numdam<\/a>.<\/li>\n<li id=\"ref-Dirichlet1829\">[Dirichlet1829] Lejeune Dirichlet, G. (1829). \u201cSur la convergence des s\u00e9ries trigonom\u00e9triques qui servent \u00e0 repr\u00e9senter une fonction arbitraire entre des limites donn\u00e9es.\u201d <i>Journal f\u00fcr die reine und angewandte Mathematik<\/i>, 4, 157\u2013169. <a href=\"https:\/\/doi.org\/10.1515\/crll.1829.4.157\">https:\/\/doi.org\/10.1515\/crll.1829.4.157<\/a>, <a href=\"https:\/\/eudml.org\/doc\/183134\">EuDML<\/a>.<\/li>\n<li id=\"ref-Lebesgue1902\">[Lebesgue1902] Lebesgue, H. (1902). \u201cInt\u00e9grale, Longueur, Aire.\u201d <i>Annali di Matematica Pura ed Applicata, Serie III<\/i>, 7, 231\u2013359. <a href=\"https:\/\/doi.org\/10.1007\/BF02420592\">https:\/\/doi.org\/10.1007\/BF02420592<\/a>. \uacf5\uac1c \uc2a4\uce94: <a href=\"https:\/\/archive.org\/details\/integralelongueu00lebe\">Internet Archive<\/a>.<\/li>\n<li id=\"ref-Axler2020\">[Axler2020] Axler, S. (2020). <i>Measure, Integration &amp; Real Analysis<\/i>. Graduate Texts in Mathematics 282. Springer. <a href=\"https:\/\/doi.org\/10.1007\/978-3-030-33143-6\">https:\/\/doi.org\/10.1007\/978-3-030-33143-6<\/a>, <a href=\"https:\/\/measure.axler.net\/\">measure.axler.net<\/a>.<\/li>\n<li id=\"ref-Vitali1905\">[Vitali1905] Vitali, G. (1905). <i>Sul problema della misura dei gruppi di punti di una retta<\/i>. Bologna: Tipografia Gamberini e Parmeggiani. <a href=\"https:\/\/books.google.com\/books?id=qRiPnQAACAAJ\">Google Books<\/a>, <a href=\"https:\/\/cs.appstate.edu\/wmcb\/Class\/5620\/ClassNotes154\/Nonmeasurable%20Set%20G%20Vitali.pdf\">https:\/\/cs.appstate.edu\/wmcb\/Class\/5620\/ClassNotes154\/Nonmeasurable%20Set%20G%20Vitali.pdf<\/a>.<\/li>\n<li id=\"ref-Caratheodory1914\">[Caratheodory1914] Carath\u00e9odory, C. (1914). \u201c\u00dcber das lineare Ma\u00df von Punktmengen\u2014eine Verallgemeinerung des L\u00e4ngenbegriffs.\u201d <i>Nachrichten von der Gesellschaft der Wissenschaften zu G\u00f6ttingen, Mathematisch-Physikalische Klasse<\/i>, 404\u2013426. <a href=\"https:\/\/eudml.org\/doc\/58921\">EuDML<\/a>.<\/li>\n<li id=\"ref-Venn1866\">[Venn1866] Venn, J. (1866). <i>The Logic of Chance: An Essay on the Foundations and Province of the Theory of Probability<\/i>. London: Macmillan. <a href=\"https:\/\/books.google.com\/books?id=4GxZAAAAcAAJ\">Google Books<\/a>.<\/li>\n<li id=\"ref-VonMises1919\">[VonMises1919] von Mises, R. (1919). \u201cGrundlagen der Wahrscheinlichkeitsrechnung.\u201d <i>Mathematische Zeitschrift<\/i>, 5, 52\u201399. <a href=\"https:\/\/doi.org\/10.1007\/BF01203155\">https:\/\/doi.org\/10.1007\/BF01203155<\/a>, <a href=\"https:\/\/eudml.org\/doc\/167538\">EuDML<\/a>.<\/li>\n<li id=\"ref-DeFinetti1931\">[DeFinetti1931] de Finetti, B. (1931). \u201cSul significato soggettivo della probabilit\u00e0.\u201d <i>Fundamenta Mathematicae<\/i>, 17, 298\u2013329. <a href=\"https:\/\/doi.org\/10.4064\/fm-17-1-298-329\">https:\/\/doi.org\/10.4064\/fm-17-1-298-329<\/a>, <a href=\"https:\/\/eudml.org\/doc\/212523\">EuDML<\/a>.<\/li>\n<li id=\"ref-Hajek2023\">[Hajek2023] H\u00e1jek, A. (2023). \u201cInterpretations of Probability.\u201d In E. N. Zalta &amp; U. Nodelman (Eds.), <i>The Stanford Encyclopedia of Philosophy<\/i> (Winter 2023 ed.). Metaphysics Research Lab, Stanford University. <a href=\"https:\/\/plato.stanford.edu\/archives\/win2023\/entries\/probability-interpret\/\">https:\/\/plato.stanford.edu\/archives\/win2023\/entries\/probability-interpret\/<\/a>.<\/li>\n<li id=\"ref-Hilbert1902\">[Hilbert1902] Hilbert, D. (1902). \u201cMathematical Problems.\u201d <i>Bulletin of the American Mathematical Society<\/i>, 8, 437\u2013479. <a href=\"https:\/\/doi.org\/10.1090\/S0002-9904-1902-00923-3\">https:\/\/doi.org\/10.1090\/S0002-9904-1902-00923-3<\/a>.<\/li>\n<li id=\"ref-Gorban2018\">[Gorban2018] Gorban, A. N. (2018). \u201cHilbert\u2019s Sixth Problem: The Endless Road to Rigour.\u201d <i>Philosophical Transactions of the Royal Society A<\/i>, 376(2118), 20170238. <a href=\"https:\/\/doi.org\/10.1098\/rsta.2017.0238\">https:\/\/doi.org\/10.1098\/rsta.2017.0238<\/a>.<\/li>\n<li id=\"ref-Kolmogorov1933\">[Kolmogorov1933] Kolmogorov, A. N. (1933). <i>Grundbegriffe der Wahrscheinlichkeitsrechnung<\/i>. Berlin: Julius Springer. <a href=\"https:\/\/doi.org\/10.1007\/978-3-642-49888-6\">https:\/\/doi.org\/10.1007\/978-3-642-49888-6<\/a>.<\/li>\n<li id=\"ref-ShaferVovk2006\">[ShaferVovk2006] Shafer, G., &amp; Vovk, V. (2006). \u201cThe Sources of Kolmogorov\u2019s <i>Grundbegriffe<\/i>.\u201d <i>Statistical Science<\/i>, 21(1), 70\u201398. <a href=\"https:\/\/doi.org\/10.1214\/088342305000000467\">https:\/\/doi.org\/10.1214\/088342305000000467<\/a>, <a href=\"https:\/\/arxiv.org\/abs\/math\/0606533\">arXiv:math\/0606533<\/a>.<\/li>\n<li id=\"ref-Howson2008\">[Howson2008] Howson, C. (2008). \u201cDe Finetti, Countable Additivity, Consistency and Coherence.\u201d <i>The British Journal for the Philosophy of Science<\/i>, 59(1), 1\u201323. <a href=\"https:\/\/doi.org\/10.1093\/bjps\/axm042\">https:\/\/doi.org\/10.1093\/bjps\/axm042<\/a>.<\/li>\n<li id=\"ref-Kallenberg2021\">[Kallenberg2021] Kallenberg, O. (2021). <i>Foundations of Modern Probability<\/i> (3rd ed.). Springer. <a href=\"https:\/\/doi.org\/10.1007\/978-3-030-61871-1\">https:\/\/doi.org\/10.1007\/978-3-030-61871-1<\/a>.<\/li>\n<li id=\"ref-Hawkins1970\">[Hawkins1970] Hawkins, T. (1970). <i>Lebesgue\u2019s Theory of Integration: Its Origins and Development<\/i>. University of Wisconsin Press. <a href=\"https:\/\/books.google.com\/books?id=ZS3vAAAAMAAJ\">Google Books<\/a>.<\/li>\n<\/ul>\n<h4>\uc800\uc791\uad8c<\/h4>\n<p>\uc774\uc2ac\ube44, 2026. designeralice\uff20daum.net.<\/p>\n<div class=\"math-abs-series-contents\">\n<p class=\"math-abs-series-menu-title\"><a href=\"..\/\">\uc218\ud559\uc801 \ucd94\uc0c1\ud654<\/a><\/p>\n<ol class=\"math-abs-series-menu-list\">\n        <!-- \ud604\uc7ac \ud398\uc774\uc9c0\uc5d0 \ud574\ub2f9\ud558\ub294 li \ud0dc\uadf8\uc5d0 class=\"math-abs-series-current-page\" \uc18d\uc131 \ucd94\uac00 --><\/p>\n<li><a href=\"..\/math-abstraction-01-essence\/\">\uc218\ud559\uc801 \ucd94\uc0c1\ud654\uc758 \uae30\ubcf8 \uac1c\ub150<\/a><\/li>\n<li><a href=\"..\/math-abstraction-02-implicit-era\/\">\ucd08\uae30 \uc218\ud559\uc5d0\uc11c\uc758 \uc554\ubb35\uc801 \ucd94\uc0c1\ud654<\/a><\/li>\n<li><a href=\"..\/math-abstraction-03-algebraic-symbols\/\">\ub300\uc218\uc801 \uae30\ud638\uc640 \uc5f0\uc0b0 \ubc95\uce59<\/a><\/li>\n<li><a href=\"..\/math-abstraction-04-axiomatic-method\/\">\uacf5\ub9ac\uc801 \ubc29\ubc95\uc5d0 \uc758\ud55c \ucd94\uc0c1\ud654<\/a><\/li>\n<li><a href=\"..\/math-abstraction-05-mathematical-structures\/\">\uc218\ud559\uc801 \uad6c\uc870\uc640 \ubc94\uc8fc\ub860<\/a><\/li>\n<li><a href=\"..\/math-abstraction-06-vector-spaces\/\">\ubca1\ud130\uacf5\uac04\uacfc \uc120\ud615 \uad6c\uc870<\/a><\/li>\n<li><a href=\"..\/math-abstraction-07-topological-spaces\/\">\uac70\ub9ac\uacf5\uac04\uacfc \uc704\uc0c1\uacf5\uac04<\/a><\/li>\n<li class=\"math-abs-series-current-page\"><a href=\"..\/math-abstraction-08-measure-spaces\/\">\uce21\ub3c4\uacf5\uac04\uacfc \ud655\ub960\uacf5\uac04<\/a><\/li>\n<li><a href=\"..\/math-abstraction-09-cognitive-process\/\">\uc218\ud559\uc801 \ucd94\uc0c1\ud654\uc758 \uc778\uc9c0\uc640 \ud559\uc2b5<\/a><\/li>\n<li><a href=\"..\/math-abstraction-10-optimal-generality\/\">\ud604\ub300 \uc218\ud559 \uc5f0\uad6c\uc5d0\uc11c\uc758 \ucd94\uc0c1\ud654<\/a><\/li>\n<li><a href=\"..\/math-abstraction-11-topography\/\">\uc218\ud559\uc801 \ucd94\uc0c1\ud654\uc758 \ubc29\ubc95\uacfc \ucca0\ud559<\/a><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><!-- class=\"math-abs-series\" --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>7\ubd80\uc5d0\uc11c \uc911\ub300\ud55c \uc9c8\ubb38 \ud558\ub098\ub97c \ub0a8\uae30\uace0 \ub9c8\ubb34\ub9ac\ud558\uc600\ub2e4. \uc2e4\uc218 \uad6c\uac04 \\([a,\\,b]\\)\uc758 \uae38\uc774 \\(b-a\\)\ub97c \ub2e8\uc21c\ud55c \uc120\ubd84\uc774 \uc544\ub2cc \ubcf5\uc7a1\ud55c \ubd80\ubd84\uc9d1\ud569\uc5d0\uae4c\uc9c0 \ud655\uc7a5\ud560 \uc218 \uc788\uc744\uae4c? \uc120\ud0dd\uacf5\ub9ac\ub97c \uac00\uc815\ud558\uba74 \uc2e4\uc218\uc758 \ube44\uac00\uce21 \ubd80\ubd84\uc9d1\ud569\uc744 \uad6c\uc131\ud560 \uc218 \uc788\ub2e4. \ub530\ub77c\uc11c \uad6c\uac04\uc758 \uae38\uc774\uc640 \uc77c\uce58\ud558\uace0, \ud3c9\ud589\uc774\ub3d9\uc5d0 \ubd88\ubcc0\uc774\uba70, \uac00\uc0b0\uac00\ubc95\uc801\uc778 \uce21\ub3c4\ub97c \uc2e4\uc218\uc758 \ubaa8\ub4e0 \ubd80\ubd84\uc9d1\ud569\uc5d0 \uc815\uc758\ud560 \uc218\ub294 \uc5c6\ub2e4. \uc5ec\uae30\uc11c \ubd88\uac00\ub2a5\ud55c \uac83\uc740 \uc544\ubb34 \uc9d1\ud569\ud568\uc218\ub098 \ub9cc\ub4dc\ub294 \uc77c\uc774 \uc544\ub2c8\ub77c, \uc6b0\ub9ac\uac00 \uae38\uc774\uc5d0 \uae30\ub300\ud558\ub294 \uc774 \uc138 \uc870\uac74\uc744 \ubaa8\ub450 \uc720\uc9c0\ud558\ub294 \uc77c\uc774\ub2e4. \uc218\ud559\uc790\ub4e4\uc740 \uc774 \ud55c\uacc4\ub97c \ubc1b\uc544\ub4e4\uc774\uace0, \uce21\uc815\ud560 \uc9d1\ud569\uc758&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":9726,"menu_order":800,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-9749","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9749","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=9749"}],"version-history":[{"count":24,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9749\/revisions"}],"predecessor-version":[{"id":10057,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9749\/revisions\/10057"}],"up":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9726"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=9749"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}