{"id":2492,"date":"2019-03-05T00:02:22","date_gmt":"2019-03-04T15:02:22","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=2492"},"modified":"2021-02-11T13:02:46","modified_gmt":"2021-02-11T04:02:46","slug":"approximation-of-measurable-functions-and-riemann-lebesgue-lemma","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/approximation-of-measurable-functions-and-riemann-lebesgue-lemma\/","title":{"rendered":"\uac00\uce21\ud568\uc218 \uadfc\uc0ac\uc640 \ub9ac\ub9cc-\ub974\ubca0\uadf8\uc758 \ubcf4\uc870\uc815\ub9ac"},"content":{"rendered":"

\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \\(\\mathcal{L}^1\\) \ub178\ub984\uc744 \uae30\uc900\uc73c\ub85c \ud588\uc744 \ub54c \uacc4\ub2e8\ud568\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \uac00\uce21\ud568\uc218\uc5d0 \uadfc\uc0ac\uc2dc\ud0a4\ub294 \ubc29\ubc95\uacfc \uc5f0\uc18d\ud568\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \\(\\mathcal{L}^1\\)\uc5d0 \uc18d\ud558\ub294 \ud568\uc218\uc5d0 \uadfc\uc0ac\ud0a4\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf4\uace0, \uadf8 \uc751\uc6a9\uc73c\ub85c\uc11c \ub9ac\ub9cc-\ub974\ubca0\uadf8\uc758 \ubcf4\uc870\uc815\ub9ac\ub97c \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n

\n

\uc815\ub9ac 1. (\uacc4\ub2e8\ud568\uc218\ub97c \uc774\uc6a9\ud55c \uac00\uce21\ud568\uc218\uc758 \uadfc\uc0ac)<\/span><\/p>\n

\\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc720\uacc4\uc778 \uac00\uce21\ud568\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc784\uc758\uc758 \\(\\epsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec \uacc4\ub2e8\ud568\uc218 \\(h\\)\uac00 \uc874\uc7ac\ud558\uc5ec
\n\\[\\int_a^b |f-h| \\, dm < \\epsilon\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.<\/p>\n<\/div>\n

\n

\uc99d\uba85<\/p>\n

\uba3c\uc800 \\(f \\ge 0\\)\uc778 \uacbd\uc6b0\ub97c \uc99d\uba85\ud558\uc790. \uadf8\ub7ec\uba74 \ub974\ubca0\uadf8 \uc801\ubd84\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec
\n\\[\\int_a^b f\\,dm = \\sup \\left\\{ \\left. \\int_a^b \\varphi \\,dm \\,\\right\\vert\\, 0 \\le \\varphi \\le f ,\\, \\varphi \\text{ is simple} \\right\\}\\]
\n\uc774\ub2e4. \\(f\\ge \\varphi\\)\uc774\ubbc0\ub85c \\(\\lvert f-\\varphi \\rvert = f-\\varphi\\)\uc774\ub2e4. \ub530\ub77c\uc11c \ub2e8\uc21c\ud568\uc218 \\(\\varphi\\)\uac00 \uc874\uc7ac\ud558\uc5ec
\n\\[\\int_a^b | f-\\varphi | \\,dm = \\int_a ^b f\\,dm – \\int_a^b \\varphi \\,dm < \\frac{\\epsilon}{2}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uc774\ub85c\uc368 \uacc4\ub2e8\ud568\uc218 \\(h\\)\ub97c \uc774\uc6a9\ud558\uc5ec \ub2e8\uc21c\ud568\uc218 \\(\\varphi\\)\uc5d0 \uadfc\uc0ac\uc2dc\ud0a4\ub294 \uc77c\ub9cc \ub0a8\uc558\ub2e4.<\/p>\n

\\(\\varphi\\)\uc758 \uc720\ud55c\uce58\uc5ed\uc744 \\[\\left\\{ a_1 ,\\, a_2 ,\\, \\cdots ,\\, a_n \\right\\}\\]\uc774\ub77c\uace0 \ud558\uba74 \\(E_i := \\varphi^{-1} (\\left\\{ a_i \\right\\})\\)\ub4e4\uc758 \ubaa8\uc784\uc740 \\([a,\\,b]\\)\uc758 \ubd84\ud560\uc744 \uc774\ub8ec\ub2e4.<\/p>\n

\uad6c\uac04\uc744 \uc774\uc6a9\ud558\uc5ec \uac01 \\(E_i\\)\ub97c \uadfc\uc0ac\uc2dc\ud0a4\uc790. \\(\\varphi\\)\uac00 \ub2e8\uc21c\ud568\uc218\uc774\ubbc0\ub85c
\n\\[M := \\sup\\left\\{ \\varphi (x) \\,\\vert\\, x\\in[a,\\,b] \\right\\} < \\infty\\]\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(i le n\\)\uc778 \uac01 \\(i\\)\uc5d0 \ub300\ud558\uc5ec \uc5f4\ub9b0\uc9d1\ud569 \\(O_i\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(E_i \\subseteq O_i\\)\uc774\uace0\n\\[m(O_i \\setminus E_i ) < \\frac{\\epsilon}{2nM}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uac01 \\(E_i\\)\uc758 \uce21\ub3c4\uac00 \uc720\ud55c\uc774\ubbc0\ub85c \\(O_i\\)\uc758 \uce21\ub3c4\ub3c4 \uc720\ud55c\uc774\uba70, \\(O_i\\)\ub294 \uc720\ud55c \uac1c\uc758 \uc5f4\ub9b0 \uad6c\uac04\uc758 \ud569\uc9d1\ud569\uc744 \uc774\uc6a9\ud558\uc5ec \uadfc\uc0ac\ub41c\ub2e4. \uc989 \uc11c\ub85c\uc18c\uc778 \uc5f4\ub9b0 \uad6c\uac04\ub4e4 \\(I_{ij}\\)\uac00 \uc874\uc7ac\ud558\uc5ec\n\\[\\begin{align}\nO_i &= \\bigcup_{j=1}^{\\infty} I_{ij}\\\\[6pt]\nm(O_i ) &= \\sum_{j=1}^{\\infty} m(I_{ij}) < \\infty\n\\end{align}\\]\n\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uc774 \ubb34\ud55c\uae09\uc218\ub294 \uc218\ub834\ud558\ubbc0\ub85c\n\\[m(O_i ) - m\\left( \\bigcup_{j=1}^{k_i} I_{ij} \\right) < \\frac{\\epsilon}{2nM}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ucca8\uc218 \\(k_i\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uac01 \\(i \\le n\\)\uc5d0 \ub300\ud558\uc5ec\n\\[G_i = \\bigcup_{j=1}^{k_i} I_{ij}\\]\n\ub77c\uace0 \ud558\uba74\n\\[\\int_a^b \\lvert \\mathbf{1}_{E_i} - \\mathbf{1}_{G_i} \\rvert \\,dm = m(E_i \\Delta G_i ) < \\frac{\\epsilon}{nM}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4.\n\\[h := \\sum_{i=1}^n \\,a_i \\mathbf{1}_{G_i}\\]\n\ub77c\uace0 \ud558\uc790. \uc774 \uacc4\ub2e8\ud568\uc218\ub294\n\\[\\int_a^b | \\varphi - h | \\,dm < \\frac{\\epsilon}{2}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a4\ubbc0\ub85c\n\\[\\int_a^b | f-h| \\,dm < \\epsilon\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

\\(f\\)\uac00 \uc74c\uc758 \uac12\uc744 \uac16\ub294 \uacbd\uc6b0\uc758 \uc99d\uba85\uc740 \uc5b4\ub835\uc9c0 \uc54a\ub2e4. \\(f^+\\)\uc640 \\(f^-\\)\ub294 \uac01\uac01 \ub2e8\uc21c\ud568\uc218 \\(h_1\\)\uacfc \\(h_2\\)\uc5d0 \uc758\ud558\uc5ec \uadfc\uc0ac\ub420 \uc218 \uc788\uc73c\uba70, \\(L^1\\)\ub178\ub984\uc758 \ucc28\uc774\uac00 \uac01\uac01 \\(\\epsilon \/ 2\\)\ubcf4\ub2e4 \uc791\ub3c4\ub85d \ub9cc\ub4e4 \uc218 \uc788\ub2e4. \\(h = h_1 – h_2\\)\ub77c\uace0 \ud558\uba74
\n\\[\\int_a^b |f-h| \\,dm
\n\\le \\int_a^b |f^+ – h_1 | \\,dm + \\int_a^b | f^- – h_2 |\\,dm < \\epsilon\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4.\n<\/span><\/p>\n<\/div>\n

\ub2e4\uc74c\uc73c\ub85c \uc5f0\uc18d\ud568\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \ub974\ubca0\uadf8 \uc801\ubd84 \uac00\ub2a5\ud55c \ud568\uc218\uc5d0 \uadfc\uc0ac\uc2dc\ucf1c \ubcf4\uc790.<\/p>\n

\n

\uc815\ub9ac 2. (\uc5f0\uc18d\ud568\uc218\ub97c \uc774\uc6a9\ud55c \\(\\mathcal{L}^1\\) \ud568\uc218\uc758 \uadfc\uc0ac)<\/span><\/p>\n

\\(f\\in\\mathcal{L}^1\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc784\uc758\uc758 \\(\\epsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec \uc5f0\uc18d\ud568\uc218 \\(g\\)\uac00 \uc874\uc7ac\ud558\uc5ec
\n\\[\\int | f-g | \\, dm < \\epsilon\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \ub354\uc6b1\uc774 \uc720\uacc4\uc778 \uad6c\uac04\uc774 \uc874\uc7ac\ud558\uc5ec \uadf8 \uad6c\uac04 \ubc16\uc5d0\uc11c \\(g\\)\uc758 \ud568\uc22b\uac12\uc774 \\(0\\)\uc774 \ub418\ub3c4\ub85d \ud560 \uc218 \uc788\ub2e4.<\/p>\n<\/div>\n

\n

\uc99d\uba85<\/p>\n

\\(f\\)\uac00 \uc720\uacc4\uc778 \uac00\uce21\ud568\uc218\uc774\uace0 \uad6c\uac04 \\([a,\\,b]\\)\uc758 \ubc14\uae65\uc5d0\uc11c \ud568\uc22b\uac12\uc774 \\(0\\)\uc774\ub77c\uace0 \ud558\uc790. \\(f=h\\)\ub77c\uace0 \ud558\uace0 \uc815\ub9ac 1\uc758 \uc99d\uba85 \uacfc\uc815\uc5d0\uc11c \uc815\uc758\ud55c \uad6c\uac04\uc744 \uc0ac\uc6a9\ud558\uc790. \uad6c\uac04 \\(I_{ij}\\)\uc758 \uc21c\uc11c\ub97c \ubc14\uafc8\uc73c\ub85c\uc368 \uc720\ud55c\uc218\uc5f4 \\((J_m )_{m\\le n} ,\\) \\(J_m = (c_m ,\\, d_m)\\)\uc774
\n\\[h = \\sum_{m=1}^{n} a_m \\mathbf{1}_{J_m}\\]
\n\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub3c4\ub85d \ud560 \uc218 \uc788\ub2e4.
\n\\[\\ell (J_m ) = (d_m – c_m ) > \\frac{\\epsilon ‘}{2}\\]
\n\uc778 \\(\\epsilon ‘ > 0\\)\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uace0 \\(\\mathbf{1}_{J_m}\\)\uc5d0 \uadfc\uc0ac\ud55c \uc5f0\uc18d\ud568\uc218 \\(g_m\\)\uc744 \uad6c\uc131\ud558\uc790. \uba3c\uc800 \\(J_m\\)\ubcf4\ub2e4 \uc0b4\uc9dd \uc791\uc740 \uad6c\uac04
\n\\[\\left( c_m + \\frac{\\epsilon ‘}{4} ,\\, d_m – \\frac{\\epsilon ‘}{4} \\right)\\]
\n\uc5d0\uc11c \ud568\uc22b\uac12\uc744 \\(g_m = 1\\)\ub85c \uc815\uc758\ud558\uace0, \\(J_m\\)\uc758 \ubc14\uae65\uc5d0\uc11c\ub294 \ud568\uc22b\uac12\uc744 \\(g_m =0\\)\uc73c\ub85c \uc815\uc758\ud55c\ub2e4. \ub610\ud55c
\n\\[J_m \\setminus \\left( c_m + \\frac{\\epsilon ‘}{4} ,\\, d_m – \\frac{\\epsilon ‘}{4} \\right)\\]
\n\uc740 \ub450 \uad6c\uac04\uc758 \ud569\uc9d1\ud569\uc774 \ub418\ub294\ub370, \uac01 \uad6c\uac04\uc5d0\uc11c \\(g\\)\uac00 \uc77c\ucc28\ud568\uc218\uac00 \ub418\ub3c4\ub85d \\(g\\)\uc758 \ud568\uc22b\uac12\uc744 \uc815\uc758\ud558\uc5ec \\(g\\)\uac00 \\(\\mathbb{R}\\)\uc5d0\uc11c \uc5f0\uc18d\uc774 \ub418\ub3c4\ub85d \ud560 \uc218 \uc788\ub2e4. \uc774\uc640 \uac19\uc774 \uad6c\uc131\ub41c \ud568\uc218 \\(g\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[\\int_a^b | \\mathbf{1}_{J_m} – g_m | \\,dm < \\frac{\\epsilon '}{2}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uac01 \\(J_m\\)\uc5d0 \ub300\ud558\uc5ec \uc774 \uc791\uc5c5\uc744 \ubc18\ubcf5\ud558\uace0,\n\\[K = \\max_{m\\le n}|a_m | ,\\quad \\epsilon ' < \\frac{\\epsilon}{nK}\\]\n\uc774\ub77c\uace0 \ud558\uc790.\n\\[g := \\sum_{m=1}^{n} a_m g_m\\]\n\uc774\ub77c\uace0 \ud558\uba74 \\(g\\)\ub294\n\\[\\int_a^b | h-g | \\,dm < \\frac{\\epsilon}{2}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.<\/p>\n

\ub2e4\uc74c\uc73c\ub85c \\(f\\in\\mathcal{L}^1 [a,\\,b]\\)\uc778 \uacbd\uc6b0\ub97c \uc99d\uba85\ud558\uc790. \uc815\ub9ac 1\uc758 \uc99d\uba85 \uacfc\uc815\uc5d0\uc11c \uc0b4\ud3b4\ubcf8 \ubc14\uc640 \uac19\uc774 \\(f \\ge 0\\)\uc774\ub77c\uace0 \uac00\uc815\ud574\ub3c4 \uc77c\ubc18\uc131\uc744 \uc783\uc9c0 \uc54a\ub294\ub2e4. \\(f_n = \\min (f,\\,n)\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(f_n\\)\uc740 \uc720\uacc4\uc778 \uac00\uce21\ud568\uc218\uc774\uace0 \\(|f_n | \\le |f|\\)\uc774\ubbc0\ub85c \uc9c0\ubc30\uc218\ub834 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(f_n \\,\\to\\,f\\)\uc774\ub2e4. \uc989 \uc801\ub2f9\ud55c \\(N\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[\\int_a^b |f-f_N | \\,dm < \\frac{\\epsilon}{2}\\]\n\uc774\ub2e4. \uc774\ub54c \uc99d\uba85\uc758 \uc55e\ubd80\ubd84\uc5d0\uc11c \ub17c\uc758\ud55c \uacb0\ub860\uc5d0 \uc758\ud558\uc5ec, \uc5f0\uc18d\ud568\uc218 \\(g\\)\uac00 \uc874\uc7ac\ud558\uc5ec\n\\[\\int_a^b |f_N - g| \\,dm\\]\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba70, \uc801\ub2f9\ud55c \uc720\ud55c \uad6c\uac04\uc758 \ubc14\uae65\uc5d0\uc11c \\(g\\)\uc758 \ud568\uc22b\uac12\uc740 \\(0\\)\uc774 \ub41c\ub2e4. \uc774\uc640 \uac19\uc774 \uc5bb\uc5b4\uc9c4 \uc5f0\uc18d\ud568\uc218 \\(g\\)\uc5d0 \ub300\ud558\uc5ec\n\\[\\int_a^b |f-g| \\,dm < \\epsilon\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

\ub05d\uc73c\ub85c \\(f\\in\\mathcal{L}^1 ( \\mathbb{R}\\)\uc774\uace0 \\(f\\ge 0\\)\uc778 \uacbd\uc6b0\ub97c \uc99d\uba85\ud558\uc790. \\(f\\)\uc758 \uc808\ub313\uac12\uc774 \ub974\ubca0\uadf8 \uc801\ubd84 \uac00\ub2a5\ud558\ubbc0\ub85c \ucda9\ubd84\ud788 \ud070 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[\\int_{|x| \\ge n} \\,f\\,dm < \\frac{\\epsilon}{3}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \ub610\ud55c \uc5f0\uc18d\ud568\uc218 \\(g\\)\uac00 \uc874\uc7ac\ud558\uc5ec\n\\[\\int_{|x| \\ge n} \\,g\\,dm < \\frac{\\epsilon}{3}\\]\n\uadf8\ub9ac\uace0\n\\[\\int_{-n}^{n} \\,|f-g| \\,dm < \\frac{\\epsilon}{3}\\]\n\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uc774 \ud568\uc218 \\(g\\)\uc5d0 \ub300\ud558\uc5ec\n\\[\\int_{\\mathbb{R}} \\,|f-g|\\,dm < \\epsilon\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4.\n<\/span><\/p>\n<\/div>\n

\ub2e4\uc74c \uc815\ub9ac\ub294 \ud478\ub9ac\uc5d0 \uae09\uc218\uc758 \uc131\uc9c8\uc744 \ubc1d\ud790 \ub54c \uc790\uc8fc \uc0ac\uc6a9\ub418\ub294 \ub4f1\uc2dd\uc744 \uc124\uba85\ud55c\ub2e4. \uc9c1\uad00\uc801\uc73c\ub85c\ub294 \uc790\uba85\ud55c \ub4f1\uc2dd\uc774\uc9c0\ub9cc \uc99d\uba85\uc740 \uaf64 \uae4c\ub2e4\ub86d\ub2e4.<\/p>\n

\n

\uc815\ub9ac 3. (Riemann-Lebesgue Lemma)<\/span><\/p>\n

\\(f\\in\\mathcal{L}^1 (\\mathbb{R})\\)\uc774\uba74
\n\\[\\begin{gather}
\n\\lim_{k\\,\\to\\,\\infty} \\int_{-\\infty}^{\\infty} f(x) \\sin kx \\,dx = 0 ,\\\\[6pt]
\n\\lim_{k\\,\\to\\,\\infty} \\int_{-\\infty}^{\\infty} f(x) \\cos kx \\,dx = 0
\n\\end{gather}\\]
\n\uc774\ub2e4.<\/p>\n<\/div>\n

\n

\uc99d\uba85<\/p>\n

\ud3b8\uc758\uc0c1 \\(\\int_{-\\infty}^{\\infty}\\)\ub97c \uac04\ub2e8\ud788 \\(\\int\\)\ub85c \ub098\ud0c0\ub0b4\uc790.
\n\\[s_k := \\int f(x) \\sin kx \\,dx\\]
\n\ub77c\uace0 \ud558\uc790. \\(x=y+\\frac{\\pi}{k}\\)\ub77c\uace0 \ud558\uba74
\n\\[\\begin{align}
\ns_k &= \\int f\\left( y+ \\frac{\\pi}{k} \\right) \\sin (ky + \\pi ) \\,dy \\\\[6pt]
\n&= – \\int f \\left( y + \\frac{\\pi}{k} \\right) \\sin (ky) \\,dy
\n\\end{align}\\]
\n\uc774\ub2e4. \\(|\\sin x | \\le 1\\)\uc774\ubbc0\ub85c
\n\\[\\begin{align}
\n\\int \\left| f(x) – f\\left( x+\\frac{\\pi}{k} \\right) \\right| dx
\n&\\ge \\left| \\int \\left( f(x) – f\\left( x + \\frac{\\pi}{k} \\right) \\right) \\sin kx \\, dx \\right| \\\\[6pt]
\n&= 2 \\left| s_k \\right|
\n\\end{align}\\]
\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c
\n\\[\\lim_{h\\,\\to\\,0} \\int | f(x) – f(x+h) | dx =0\\]
\n\uc784\uc744 \uc99d\uba85\ud558\uba74 \ucda9\ubd84\ud558\ub2e4.<\/p>\n

\\(\\epsilon > 0\\)\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uc815\ub9ac 2\uc5d0 \uc758\ud558\uc5ec \uc5f0\uc18d\ud568\uc218 \\(g\\)\uac00 \uc874\uc7ac\ud558\uc5ec
\n\\[\\int | f-g | \\,dm < \\frac{\\epsilon}{3}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a4\uace0 \\([a,\\,b]\\)\uc758 \ubc14\uae65\uc5d0\uc11c \\(g\\)\uc758 \ud568\uc22b\uac12\uc740 \\(0\\)\uc774 \ub41c\ub2e4. \\(|h| < 1\\)\uc778 \\(h\\)\uc5d0 \ub300\ud558\uc5ec\n\\[g_h (x) = g(x+h)\\]\n\ub85c \uc815\uc758\ub41c \ud568\uc218 \\(g_h\\)\ub294 \uc5f0\uc18d\uc774\uba70 \\([a-1,\\,b+1]\\)\uc758 \ubc14\uae65\uc5d0\uc11c \\(g_h\\)\uc758 \ud568\uc22b\uac12\uc740 \\(0\\)\uc774 \ub41c\ub2e4. \ub610\ud55c \ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4.\n\\[\\begin{align}\n\\int | f(x+h) &- f(x)| \\,dm\n\\le \\int | f(x+h) -g(x+h)| \\,dm \\\\[6pt]\n&+ \\int |g(x+h) - g(x)| \\,dm + \\int | g(x) - f(x) |\\,dm .\n\\end{align}\\]\n\uc6b0\ubcc0\uc758 \uccab \uc801\ubd84\uacfc \uc138 \ubc88\uc9f8 \uc801\ubd84\uc758 \uac12\uc740 \ubaa8\ub450 \\(\\epsilon \/ 3\\)\ubcf4\ub2e4 \uc791\uc73c\uba70, \\(g\\)\uac00 \uc5f0\uc18d\ud568\uc218\uc774\ubbc0\ub85c \ucda9\ubd84\ud788 \uc791\uc740 \\(\\delta > 0\\)\uc5d0 \ub300\ud558\uc5ec \\(|h| < \\delta\\)\uc77c \ub54c \uc6b0\ubcc0\uc758 \ub450 \ubc88\uc9f8 \uc801\ubd84\uc758 \uac12\uc740 \\(\\epsilon \/ (3(b-a+2))\\)\ubcf4\ub2e4 \uc791\ub2e4. \\([a-1 ,\\, b+1]\\)\uc758 \ubc14\uae65\uc5d0\uc11c \\(g\\)\uc758 \ud568\uc22b\uac12\uc774 \\(0\\)\uc774\ubbc0\ub85c \ub450 \ubc88\uc9f8 \uc801\ubd84\uc758 \uac12\uc740 \\(\\epsilon \/ 3\\)\ubcf4\ub2e4\ub3c4 \uc791\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(|h| < \\delta\\)\uc77c \ub54c\n\\[\\int | f(x+h) - f(x) | \\,dm < \\epsilon\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

\uc9c0\uae08\uae4c\uc9c0\uc758 \uc99d\uba85 \uacfc\uc815\uc5d0\uc11c \\(\\sin\\)\uc744 \\(\\cos\\)\uc73c\ub85c \ubc14\uafb8\uba74 \uc815\ub9ac\uc758 \ub450 \ubc88\uc9f8 \ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.
\n<\/span><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \\(\\mathcal{L}^1\\) \ub178\ub984\uc744 \uae30\uc900\uc73c\ub85c \ud588\uc744 \ub54c \uacc4\ub2e8\ud568\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \uac00\uce21\ud568\uc218\uc5d0 \uadfc\uc0ac\uc2dc\ud0a4\ub294 \ubc29\ubc95\uacfc \uc5f0\uc18d\ud568\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \\(\\mathcal{L}^1\\)\uc5d0 \uc18d\ud558\ub294 \ud568\uc218\uc5d0 \uadfc\uc0ac\ud0a4\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf4\uace0, \uadf8 \uc751\uc6a9\uc73c\ub85c\uc11c \ub9ac\ub9cc-\ub974\ubca0\uadf8\uc758 \ubcf4\uc870\uc815\ub9ac\ub97c \uc0b4\ud3b4\ubcf8\ub2e4. \uc815\ub9ac 1. (\uacc4\ub2e8\ud568\uc218\ub97c \uc774\uc6a9\ud55c \uac00\uce21\ud568\uc218\uc758 \uadfc\uc0ac) \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc720\uacc4\uc778 \uac00\uce21\ud568\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc784\uc758\uc758 \\(\\epsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec \uacc4\ub2e8\ud568\uc218 \\(h\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(\\int_a^b |f-h| \\, dm < \\epsilon\\) \uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uc99d\uba85 \uba3c\uc800 \\(f \\ge 0\\)\uc778 \uacbd\uc6b0\ub97c \uc99d\uba85\ud558\uc790. \uadf8\ub7ec\uba74 \ub974\ubca0\uadf8 \uc801\ubd84\uc758 \uc815\uc758\uc5d0…\n<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"categories":[51],"tags":[],"class_list":["post-2492","post","type-post","status-publish","format-standard","hentry","category-real-analysis"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2492","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"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=2492"}],"version-history":[{"count":20,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2492\/revisions"}],"predecessor-version":[{"id":6137,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2492\/revisions\/6137"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=2492"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=2492"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=2492"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}