{"id":25435,"date":"2021-03-12T00:00:03","date_gmt":"2021-03-12T00:00:03","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=25435"},"modified":"2024-06-05T04:20:12","modified_gmt":"2024-06-05T04:20:12","slug":"%e4%bb%80%e4%b9%88%e6%98%af%e8%a5%bf%e6%a0%bc%e7%8e%9b%e4%bb%a3%e6%95%b0%ef%bc%9f%e5%ae%9a%e4%b9%89%e5%92%8c%e7%a4%ba%e4%be%8b","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/zh\/%e4%bb%80%e4%b9%88%e6%98%af%e8%a5%bf%e6%a0%bc%e7%8e%9b%e4%bb%a3%e6%95%b0%ef%bc%9f%e5%ae%9a%e4%b9%89%e5%92%8c%e7%a4%ba%e4%be%8b\/","title":{"rendered":"\u4ec0\u4e48\u662f\u897f\u683c\u739b\u4ee3\u6570\uff1f\u5b9a\u4e49\u548c\u793a\u4f8b"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<p><center><\/p>\n<h1>\u4ec0\u4e48\u662f\u03c3-\u4ee3\u6570\uff1f\u5b9a\u4e49\u548c\u793a\u4f8b<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>\u6458\u8981<\/strong><br \/><em>\u8fd9\u8282\u8bfe\u63a2\u8ba8\u4e86\u03c3-\u4ee3\u6570\u5728\u6982\u7387\u8bba\u4e2d\u7684\u91cd\u8981\u6027\u3002\u03c3-\u4ee3\u6570\u662f\u4e00\u4e2a\u5305\u542b\u6837\u672c\u7a7a\u95f4\u4e2d\u6240\u6709\u53ef\u6d4b\u4e8b\u4ef6\u7684\u7ed3\u6784\uff0c\u5b83\u5141\u8bb8\u5b9a\u4e49\u6982\u7387\u6d4b\u5ea6\u3002\u901a\u8fc7\u5b9e\u9645\u4f8b\u5b50\uff0c\u5982\u629b\u786c\u5e01\u548c\u7535\u5b50\u8bbe\u5907\u7684\u5bff\u547d\uff0c\u6211\u4eec\u89e3\u91ca\u4e86\u03c3-\u4ee3\u6570\u662f\u5982\u4f55\u4ece\u6837\u672c\u7a7a\u95f4\u7684\u90e8\u5206\u6784\u5efa\u800c\u6765\u7684\u3002\u8fd8\u4ecb\u7ecd\u4e86\u4e0e\u8fde\u7eed\u6837\u672c\u7a7a\u95f4\u76f8\u5173\u7684Borel \u03c3-\u4ee3\u6570\uff0c\u5e76\u89e3\u91ca\u4e86\u5176Borelian\u4e8b\u4ef6\u3002<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>\u5b66\u4e60\u76ee\u6807\uff1a<\/strong><br \/>\n\u5b8c\u6210\u672c\u8bfe\u540e\uff0c\u5b66\u751f\u5c06\u80fd\u591f\uff1a\n<\/p>\n<ol>\n<li><strong>\u7406\u89e3<\/strong>\u03c3-\u4ee3\u6570\u7684\u5b9a\u4e49\u548c\u7279\u5f81\uff0c\u4f5c\u4e3a\u4e00\u79cd\u6570\u5b66\u7ed3\u6784\uff0c\u7528\u4e8e\u5b9a\u4e49\u6982\u7387\u6d4b\u5ea6\u3002<\/li>\n<li><strong>\u8bc6\u522b<\/strong>\u6784\u6210\u03c3-\u4ee3\u6570\u7684\u5143\u7d20\u53ca\u5176\u4e0e\u6837\u672c\u7a7a\u95f4\u4e2d\u53ef\u6d4b\u4e8b\u4ef6\u7684\u5173\u7cfb\u3002<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>\u76ee\u5f55<\/strong><br \/>\n<a href=\"#1\">\u03c3-\u4ee3\u6570\u7684\u5b9a\u4e49<\/a><br \/>\n<a href=\"#2\">\u629b\u786c\u5e01\u4e2d\u7684\u03c3-\u4ee3\u6570<\/a><br \/>\n<a href=\"#3\">\u8fde\u7eed\u60c5\u51b5\u4e0b\u7684\u03c3-\u4ee3\u6570<\/a><br \/>\n<center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/cBnDYSiuyfE\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>\n<\/div>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=cBnDYSiuyfE&amp;t=121s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">\u53ef\u6d4b\u4e8b\u4ef6\u51fa\u73b0\u5728\u6982\u7387\u7a7a\u95f4\u4e2d\uff0c\u901a\u8fc7\u03c3-\u4ee3\u6570\u5f97\u5230\u3002<\/span><\/strong><\/a>\u901a\u8fc7\u8fd9\u4e2a\u6982\u5ff5\uff0c\u6211\u4eec\u5c06\u6700\u521d\u76f4\u89c2\u7684\u6982\u5ff5\u8f6c\u5316\u4e3a\u4e00\u79cd\u5f62\u5f0f\u5316\u7684\u6570\u5b66\u7ed3\u6784\uff0c\u4ece\u800c\u5b9a\u4e49\u6982\u7387\u6d4b\u5ea6\u3002<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>\u03c3-\u4ee3\u6570\u7684\u5b9a\u4e49<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=cBnDYSiuyfE&amp;t=209s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">\u03c3-\u4ee3\u6570\uff08\u6216\u03c3-\u4ee3\u6570\uff09\u662f\u4e00\u4e2a\u5305\u542b\u6837\u672c\u7a7a\u95f4\u4e2d\u6240\u6709\u53ef\u6d4b\u4e8b\u4ef6\u7684\u7ed3\u6784\u3002<\/span><\/strong><\/a>\u5982\u679c\u6ee1\u8db3\u4ee5\u4e0b\u6761\u4ef6\uff0c\u90a3\u4e48\u5bf9\u4e8e\u4e00\u4e2a\u6837\u672c\u7a7a\u95f4<span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span>\uff0c\u6211\u4eec\u79f0\u5176\u4e3a\u03c3-\u4ee3\u6570\u5bf9<span class=\"katex-eq\" data-katex-display=\"false\">\\Sigma_{\\Omega} = (\\Omega, \\mathcal{A}_{\\Omega})<\/span>\uff1a\n<\/p>\n<ol style=\"text-align: justify; color: #000000;\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\emptyset,\\Omega \\in \\mathcal{A}_\\Omega<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\left(E \\in \\mathcal{A}_\\Omega \\right) \\rightarrow (E^c = \\Omega\\setminus E \\in \\mathcal{A}_\\Omega)<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\left(E_1, E_2 \\in \\mathcal{A}_\\Omega \\right) \\rightarrow (E_1 \\cup E_2 \\in \\mathcal{A}_\\Omega)<\/span><\/li>\n<\/ol>\n<p style=\"text-align: justify; color: #000000;\">\u6240\u6709<span class=\"katex-eq\" data-katex-display=\"false\">E\\in\\mathcal{A}_\\Omega<\/span>\u4e2d\u7684\u5bf9\u8c61\u79f0\u4e3a<strong>\u6837\u672c\u7a7a\u95f4<span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span>\u7684\u4e8b\u4ef6<\/strong>\u3002<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>\u629b\u786c\u5e01\u4e2d\u7684\u03c3-\u4ee3\u6570<\/h2>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: justify; color: #008000; background-color: #dddddd;\"><strong>\u793a\u4f8b 1<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: justify; color: #000000; background-color: #ffffff;\"><a href=\"https:\/\/www.youtube.com\/watch?v=cBnDYSiuyfE&amp;t=321s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">\u5bf9\u4e8e\u629b\u4e00\u679a\u786c\u5e01\u7684\u60c5\u51b5\uff0c<\/span><\/strong><\/a>\u03c3-\u4ee3\u6570\u7531<span class=\"katex-eq\" data-katex-display=\"false\">\\Sigma_{1m}=(\\Omega_{1m}, \\mathcal{A}_{1m})<\/span>\u7ed9\u51fa\uff0c\u5176\u4e2d<\/p>\n<ul>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{1m}= \\{C,S\\}<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}_{1m}= \\{\\emptyset,\\{C\\},\\{S\\}, \\Omega_{1m}\\}<\/span><\/li>\n<\/ul>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}_{1m}<\/span>\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u8868\u793a\u4e00\u4e2a\u4e8b\u4ef6\uff1a<\/p>\n<ul>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\emptyset<\/span> \u00ab\u4e0d\u51fa\u73b0\u6b63\u9762\u6216\u53cd\u9762\u00bb\uff08\u4e0d\u53ef\u80fd\u4e8b\u4ef6\uff09\u3002<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{C\\}<\/span> \u00ab\u51fa\u73b0\u6b63\u9762\u00bb\u3002<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{S\\}<\/span> \u00ab\u51fa\u73b0\u53cd\u9762\u00bb\u3002<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{1m}= \\{C,S\\}<\/span> \u00ab\u6b63\u53cd\u9762\u90fd\u6709\u53ef\u80fd\u00bb\uff08\u5fc5\u7136\u4e8b\u4ef6\uff09\u3002<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: justify; color: #008000; background-color: #dddddd;\"><strong>\u793a\u4f8b 2<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: justify; color: #000000; background-color: #ffffff;\"><a href=\"https:\/\/www.youtube.com\/watch?v=cBnDYSiuyfE&amp;t=558s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">\u5982\u679c\u6211\u4eec\u629b\u4e24\u679a\u786c\u5e01\uff0c<\/span><\/strong><\/a>\u53ef\u4ee5\u901a\u8fc7<span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{2m}<\/span>\u7684\u90e8\u5206\u5f97\u5230\u4e00\u4e2a\u53ef\u80fd\u7684\u03c3-\u4ee3\u6570<span class=\"katex-eq\" data-katex-display=\"false\">\\Sigma_{2m}=(\\Omega_{2m}, \\mathcal{A}_{2m})<\/span>\u3002\u6211\u4eec\u6709\u4ee5\u4e0b\u5185\u5bb9\uff1a<\/p>\n<ul>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{2m}= \\{(C,C);(C,S); (S,C); (S,S)\\}<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}_{2m}=\\mathcal{P}(\\Omega_{2m}) = \\cdots<\/span>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\cdots = \\{\\emptyset; \\{(C,C)\\}; \\left\\{(C,S)\\}; \\{(S,C)\\}; \\{(S,S)\\}; \\cdots \\right. <\/span><span class=\"katex-eq\" data-katex-display=\"false\">\\cdots; \\{(C,C);(C,S)\\};\\{(C,C);(S,C)\\};\\{(C,C);(S,S)\\};\\cdots<\/span><span class=\"katex-eq\" data-katex-display=\"false\">\\cdots; \\{(C,S);(S,C)\\};\\{(C,S);(S,S)\\};\\{(S,C);(S,S)\\};\\cdots<\/span><span class=\"katex-eq\" data-katex-display=\"false\">\\cdots; \\{(C,C);(C,S);(S,C)\\};\\{(C,C);(C,S);(S,S)\\}\\cdots<\/span>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\left.\\cdots; \\{(C,C);(S,C);(S,S)\\}; \\{(C,S);(S,C);(S,S)\\}; \\Omega_{2m}\\right\\}<\/span><\/li>\n<\/ul>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}_{2m}<\/span>\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u8868\u793a<span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{2m}<\/span>\u7684\u4e00\u4e2a\u4e8b\u4ef6\u3002\u4ee5\u4e0b\u662f\u5176\u4e2d\u4e00\u4e9b\u4e8b\u4ef6\u7684\u540d\u79f0\uff1a<\/p>\n<ul>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\emptyset<\/span> \u00ab\u6ca1\u6709\u7ed3\u679c\u00bb\uff08\u4e0d\u53ef\u80fd\u4e8b\u4ef6\uff09\u3002<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{(C,C)\\}<\/span> \u00ab\u4e24\u6b21\u90fd\u662f\u6b63\u9762\u00bb\u3002<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{(C,S)\\}<\/span> \u00ab\u7b2c\u4e00\u6b21\u662f\u6b63\u9762\uff0c\u7b2c\u4e8c\u6b21\u662f\u53cd\u9762\u00bb\u3002<br \/>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{(C,C);(C,S)\\}<\/span> \u00ab\u7b2c\u4e00\u6b21\u662f\u6b63\u9762\uff0c\u7b2c\u4e8c\u6b21\u4efb\u610f\u00bb\u3002<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{(C,C);(S,C)\\}<\/span> \u00ab\u7b2c\u4e00\u6b21\u4efb\u610f\uff0c\u7b2c\u4e8c\u6b21\u662f\u6b63\u9762\u00bb\u3002<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{(C,C);(S,S)\\}<\/span> \u00ab\u4e24\u6b21\u7ed3\u679c\u76f8\u540c\u00bb\u3002<br \/>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{(C,C);(C,S);(S,S)\\}<\/span> \u00ab\u5982\u679c\u7b2c\u4e00\u6b21\u662f\u53cd\u9762\uff0c\u7b2c\u4e8c\u6b21\u4e5f\u662f\u53cd\u9762\uff1b\u5426\u5219\u7b2c\u4e8c\u6b21\u4efb\u610f\u00bb\u3002<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{(C,C);(S,C);(S,S)\\}<\/span> \u00ab\u5982\u679c\u7b2c\u4e00\u6b21\u662f\u6b63\u9762\uff0c\u7b2c\u4e8c\u6b21\u4e5f\u662f\u6b63\u9762\uff1b\u5426\u5219\u7b2c\u4e8c\u6b21\u4efb\u610f\u00bb\u3002<br \/>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{2m}<\/span> \u00ab\u4efb\u610f\u53ef\u80fd\u7684\u7ed3\u679c\u00bb\uff08\u5fc5\u7136\u4e8b\u4ef6\uff09\u3002<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"3\"><\/a><\/p>\n<h2>\u8fde\u7eed\u60c5\u51b5\u4e0b\u7684\u03c3-\u4ee3\u6570<\/h2>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: justify; color: #008000; background-color: #dddddd;\"><strong>\u793a\u4f8b 3<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: justify; color: #000000; background-color: #ffffff;\"><a href=\"https:\/\/www.youtube.com\/watch?v=cBnDYSiuyfE&amp;t=1265s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">\u5bf9\u4e8e\u7535\u5b50\u8bbe\u5907\u7684\u5bff\u547d\uff08\u4ee5\u5c0f\u65f6\u4e3a\u5355\u4f4d\uff09\uff0c\u5b83\u53ef\u80fd\u5728\u4efb\u4f55\u65f6\u523b\u635f\u574f\uff0c<\/span><\/strong><\/a>\u03c3-\u4ee3\u6570<span class=\"katex-eq\" data-katex-display=\"false\">\\Sigma_e = (\\Omega_e, \\mathcal{A}_e)<\/span>\u7ed9\u51fa\u5982\u4e0b\uff1a<\/p>\n<ul>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_e = [0, \\infty[ <\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}_e = \\{I \\; | \\; I \\subseteq \\Omega_e \\}<\/span><\/li>\n<\/ul>\n<p>\u56e0\u6b64\uff0c<span class=\"katex-eq\" data-katex-display=\"false\">I_t = ]0,t[\\in\\mathcal{A}_e <\/span>\u53ef\u4ee5\u89e3\u91ca\u4e3a\u00bb\u7535\u5b50\u8bbe\u5907\u8fde\u7eed\u8fd0\u884c[t[\/latex]\u5c0f\u65f6\u76f4\u5230\u635f\u574f\u00bb\u3002<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">\u4e0e\u8fde\u7eed\u6837\u672c\u7a7a\u95f4\u76f8\u5173\u7684\u6982\u7387\u03c3-\u4ee3\u6570\u4e5f\u88ab\u79f0\u4e3a<strong>Borel \u03c3-\u4ee3\u6570<\/strong>\uff0c\u5176\u4e8b\u4ef6\u79f0\u4e3aBorelian\u4e8b\u4ef6\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u4ec0\u4e48\u662f\u03c3-\u4ee3\u6570\uff1f\u5b9a\u4e49\u548c\u793a\u4f8b \u6458\u8981\u8fd9\u8282\u8bfe\u63a2\u8ba8\u4e86\u03c3-\u4ee3\u6570\u5728\u6982\u7387\u8bba\u4e2d\u7684\u91cd\u8981\u6027\u3002\u03c3-\u4ee3\u6570\u662f\u4e00\u4e2a\u5305\u542b\u6837\u672c\u7a7a\u95f4\u4e2d\u6240\u6709\u53ef\u6d4b\u4e8b\u4ef6\u7684\u7ed3\u6784\uff0c\u5b83\u5141\u8bb8\u5b9a\u4e49\u6982\u7387\u6d4b\u5ea6\u3002\u901a\u8fc7\u5b9e\u9645\u4f8b\u5b50\uff0c\u5982\u629b\u786c\u5e01\u548c\u7535\u5b50\u8bbe\u5907\u7684\u5bff\u547d\uff0c\u6211\u4eec\u89e3\u91ca\u4e86\u03c3-\u4ee3\u6570\u662f\u5982\u4f55\u4ece\u6837\u672c\u7a7a\u95f4\u7684\u90e8\u5206\u6784\u5efa\u800c\u6765\u7684\u3002\u8fd8\u4ecb\u7ecd\u4e86\u4e0e\u8fde\u7eed\u6837\u672c\u7a7a\u95f4\u76f8\u5173\u7684Borel \u03c3-\u4ee3\u6570\uff0c\u5e76\u89e3\u91ca\u4e86\u5176Borelian\u4e8b\u4ef6\u3002 \u5b66\u4e60\u76ee\u6807\uff1a \u5b8c\u6210\u672c\u8bfe\u540e\uff0c\u5b66\u751f\u5c06\u80fd\u591f\uff1a 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