{"id":39709,"date":"2011-09-27T19:09:22","date_gmt":"2011-09-27T11:09:22","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=39709"},"modified":"2021-10-06T16:22:46","modified_gmt":"2021-10-06T08:22:46","slug":"%e6%a9%9f%e7%8e%87%e7%a9%ba%e9%96%93%ef%bc%883%ef%bc%89%e6%a9%9f%e7%8e%87%e7%a9%ba%e9%96%93%ef%bc%88probability-space-3-probability-space%ef%bc%89","status":"publish","type":"post","link":"http:\/\/localhost\/%e6%a9%9f%e7%8e%87%e7%a9%ba%e9%96%93%ef%bc%883%ef%bc%89%e6%a9%9f%e7%8e%87%e7%a9%ba%e9%96%93%ef%bc%88probability-space-3-probability-space%ef%bc%89\/","title":{"rendered":"\u6a5f\u7387\u7a7a\u9593\uff083\uff09\u6a5f\u7387\u7a7a\u9593\uff08Probability space-3. Probability space\uff09"},"content":{"rendered":"

\u6a5f\u7387\u7a7a\u9593<\/span>\uff08<\/span>3\uff09\u6a5f\u7387\u7a7a\u9593<\/span>\uff08Probability space-3. Probability space\uff09<\/span>
\n\u570b\u7acb\u9ad8\u96c4\u5927\u5b78\u61c9\u7528\u6578\u5b78\u7cfb\u9ec3\u6587\u748b\u6559\u6388\/\u570b\u7acb\u9ad8\u96c4\u5927\u5b78\u61c9\u7528\u6578\u5b78\u7cfb\u9ec3\u6587\u748b\u6559\u6388\u8cac\u4efb\u7de8\u8f2f<\/span><\/strong><\/p>\n

\u9023\u7d50\uff1a\u6a5f\u7387\u7a7a\u9593\uff082\uff09\u6a5f\u7387\u7684\u610f\u7fa9<\/a><\/p>\n

\u6458\u8981\uff1a\u6a5f\u7387\u7a7a\u9593\u662f\u6a5f\u7387\u8ad6\u7684\u57fa\u790e\uff0c\u672c\u7bc7\u5f9e\u96c6\u5408\u8ad6\u51fa\u767c\uff0c\u4ecb\u7d39Kolmogorov\u7684\u73fe\u4ee3\u516c\u8a2d\u89c0\u9ede\uff0c\u4e26\u4ee5\u6b64\u5b9a\u7fa9\u6a5f\u7387\u51fd\u6578(probability function)\uff0c\u4e26\u689d\u5217\u51fa\u6a5f\u7387\u51fd\u6578\u7684\u4e00\u4e9b\u6027\u8cea\u3002<\/strong><\/p>\n

\u773e\u6240\u5468\u77e5\uff0c\u6578\u5b78\u4e2d\u5404\u9818\u57df\u7684\u8d77\u6e90\uff0c\u901a\u5e38\u90fd\u662f\u59cb\u65bc\u89e3\u6c7a\u5be6\u969b\u7684\u554f\u984c\uff0c\u6b64\u6642\u4e0d\u8ad6\u5c08\u696d\u6216\u696d\u9918\uff0c\u5f88\u591a\u4eba\u90fd\u53ef\u53c3\u8207\u63a2\u8a0e\u3002\u800c\u5f8c\u9010\u6f38\u6df1\u5165\uff0c\u5c31\u53ea\u6709\u5c11\u6578\u5c08\u696d\u4eba\u58eb\u80fd\u7406\u89e3\u5176\u4e2d\u7684\u5167\u6db5\u4e86\u3002<\/p>\n

\u4ee5\u5e7e\u4f55\u5b78\u70ba\u4f8b\uff0c\u5176\u767c\u6e90\u662f\u59cb\u65bc\u5c3c\u7f85\u6cb3\u6c3e\u6feb\u5f8c\u7684\u6e2c\u91cf\u554f\u984c\u3002\u5e73\u9762\u5e7e\u4f55\u5b78\u4e2d\u7684\u8af8\u591a\u7f8e\u5999\u7d50\u679c\uff0c\u5169\u5343\u4e09\u767e\u591a\u5e74\u524d\uff0c\u5c31\u5df2\u88ab\u6b50\u5e7e\u91cc\u5f97(Euclid\uff0c\u7d04\u897f\u5143\u524d375-300\u5e74)\uff0c\u6536\u9304\u65bc\u5176\u5e7e\u4f55\u539f\u672c(Elements)\u4e00\u66f8\u4e2d\u3002\u7576\u5e74\u7562\u9054\u54e5\u62c9\u65af(Pythagoras\uff0c\u7d04\u897f\u5143\u524d580-500\u5e74)\u7b49\u8457\u540d\u6578\u5b78\u5bb6\u6240\u63a2\u8a0e\u7684\u554f\u984c\uff0c\u4e0d\u904e\u662f\u4eca\u65e5\u4e2d\u5b78\u6578\u5b78\u8ab2\u7a0b\u7684\u5167\u5bb9\u3002\u800c\u4e00\u822c\u4eba\u82e5\u7ffb\u95b1\u5927\u5b78\u6578\u5b78\u7cfb\u7684\u5e7e\u4f55\u5b78\u6559\u79d1\u66f8\uff0c\u53ef\u80fd\u4e0d\u6703\u611f\u89ba\u9019\u662f\u5728\u8b1b\u5e7e\u4f55\u7684\u66f8\u3002<\/p>\n

\u6578\u5b78\u7684\u767c\u5c55\uff0c\u901a\u5e38\u6709\u4e09\u500b\u968e\u6bb5\u3002\u7b2c\u4e00\u968e\u6bb5\u662f\u89e3\u6c7a\u5be6\u969b\u554f\u984c\uff0c\u6b64\u6642\u4eba\u5011\u53ea\u6c42\u5c07\u9019\u4e9b\u554f\u984c\u7684\u89e3\u7b54\u627e\u51fa\u4f86\u3002\u4f46\u9010\u6f38\u5730\uff0c\u6709\u4e9b\u4eba\u5728\u89e3\u6c7a\u554f\u984c\u4e4b\u9918\uff0c\u4e5f\u9032\u4e00\u6b65\u63a2\u8a0e\u9019\u4e9b\u554f\u984c\u7684\u672c\u8cea\uff0c\u4e5f\u5c31\u662f\u5efa\u7acb\u8f03\u4e00\u822c\u7684\u7cfb\u7d71\uff0c\u800c\u5c07\u539f\u6709\u7684\u554f\u984c\u7576\u4f5c\u6b64\u7cfb\u7d71\u4e0b\u4e4b\u7279\u4f8b\u3002\u4ee5\u6392\u5217\u7d44\u5408\u89e3\u6c7a\u4e00\u822c\u7684\u6a5f\u7387\u554f\u984c\uff0c\u53ef\u8996\u70ba\u6a5f\u7387\u767c\u5c55\u7684\u7b2c\u4e8c\u968e\u6bb5\u3002\u4f46\u82e5\u672a\u9032\u5165\u7b2c\u4e09\u968e\u6bb5\uff0c\u6a5f\u7387\u662f\u7121\u6cd5\u6210\u70ba\u6578\u5b78\u4e2d\u4e4b\u4e00\u4e3b\u8981\u9818\u57df\u3002<\/p>\n

\u7531\u65bc\u8d77\u6e90\u8f03\u665a\uff0c\u76f4\u523020\u4e16\u7d00\uff0c\u65bc1933\u5e74\uff0c\u5e74\u8f15\u7684\u4fc4\u570b\u6578\u5b78\u5bb6\u67ef\u83ab\u679c\u6d1b\u592b(Andrey N. Kolmogorov\uff0c1903-1987)\uff0c\u624d\u4ee5\u516c\u7406\u5316\u7684\u65b9\u5f0f\u4f86\u5b9a\u7fa9\u6a5f\u7387\u3002\u7d93\u7531\u4ed6\u6240\u7d66\u7684\u5e7e\u500b\u516c\u7406(axiom)\uff0c\u53ef\u63a8\u5c0e\u51fa\u8a31\u591a\u7d50\u679c\uff0c\u56e0\u800c\u53ef\u5c0d\u6211\u5011\u6240\u63a2\u8a0e\u7684\u4e00\u4e9b\u5be6\u969b\u554f\u984c\uff0c\u6709\u66f4\u6e05\u6670\u7684\u6982\u5ff5\u3002\u7576\u7136\uff0c\u4e5f\u53ef\u53ea\u662f\u63a2\u8a0e\uff0c\u6240\u8b02\u88ab\u7a31\u70ba\u6a5f\u7387\u800c\u4e0d\u7528\u7ba1\u9019\u4e9b\u984c\u6750\u8207\u4efb\u4f55\u5be6\u969b\u7684\u554f\u984c\u6709\u95dc\u806f\u3002<\/p>\n

\u6a5f\u7387\uff0c\u5f80\u6614\u66fe\u88ab\u7a31\u70ba\u6216\u7136\u7387\u3001\u6982\u7387\u7b49\u3002\u5982\u4f55\u80fd\u4ee5\u66f4\u4e00\u822c\u7684\u65b9\u5f0f\u5f15\u9032\uff0c\u4e14\u53ef\u5305\u542b\u4e0a\u7bc0\u4e2d\u6240\u7d66\u7684\u4e09\u7a2e\u5c0d\u6a5f\u7387\u7684\u89e3\u91cb\uff0c\u53c8\u5177\u5145\u5206\u7684\u5f48\u6027\uff0c\u80fd\u8b93\u6211\u5011\u63a8\u5c0e\u51fa\u4e00\u5957\u5b8c\u6574\u7684\u7406\u8ad6?\u9996\u5148\u7531\u4e0a\u4e00\u7bc0\u4e2d\uff0c\u4e09\u7a2e\u5c0d\u6a5f\u7387\u7684\u89e3\u91cb\uff0c\u5f97\u5230\u555f\u793a\uff0c\u5b9a\u7fa9\u4e2d\u5fc5\u9808\u5305\u542b\u4e0b\u8ff0\u4e09\u57fa\u672c\u8981\u4ef6\uff1a<\/p>\n

    \n
  1. \u4e00\u975e\u7a7a\u7684\u96c6\u5408 $$\\Omega$$\uff0c\u7a31\u70ba\u6a23\u672c\u7a7a\u9593<\/li>\n
  2. \u4e00 $$\\Omega$$ \u7684\u4e00\u4e9b\u5b50\u96c6\u5408\u5f62\u6210\u7684\u96c6\u5408 $$\\mathcal{F}$$\uff0c$$\\mathcal{F}$$ \u4e2d\u7684\u5143\u7d20\u7a31\u70ba\u4e8b\u4ef6<\/li>\n
  3. \u5c0d\u6bcf\u4e00\u500b\u4e8b\u4ef6 $$A\\in\\mathcal{F}$$\uff0c\u7d66\u5b9a\u4e00\u500b\u5be6\u6578 $$P(A)$$\uff0c\u7a31\u70ba $$A$$ \u4e4b\u6a5f\u7387<\/li>\n<\/ol>\n

    \u4e0a\u8ff0 $$P$$ \u5373\u70ba\u4e00\u500b\u5b9a\u7fa9\u5728 $$\\mathcal{F}$$ \u4e2d\u7684\u53d6\u5be6\u503c\u4e4b\u51fd\u6578\u3002\u7531\u65bc\u5b9a\u7fa9\u57df $$\\mathcal{F}$$ \u4e2d\u7684\u6bcf\u4e00\u5143\u7d20\u7686\u70ba{\u96c6\u5408\u51fd\u6578}(set function)\u3002\u6a23\u672c\u7a7a\u9593\u3001 \u4e8b\u4ef6\u53ca\u6a5f\u7387\uff0c\u7686\u53ef\u6709\u5982\u4e0a\u7bc0\u4e2d\u7684\u89e3\u91cb\u3002\u5373 $$\\Omega$$ \u53ef\u8996\u70ba\u4e00\u89c0\u6e2c\u4e4b\u6240\u6709\u53ef\u80fd\u7684\u7d50\u679c\u4e4b\u96c6\u5408; \u4e00\u4e8b\u4ef6 $$A$$ \u7a31\u70ba\u767c\u751f\uff0c\u5982\u679c\u89c0\u6e2c\u7684\u7d50\u679c\u70ba $$A$$ \u4e2d\u7684\u5143\u7d20; \u800c $$P(A)$$ \u53ef\u8996\u70ba\u4e8b\u4ef6 $$A$$\uff0c\u4e0d\u8ad6\u5728\u90a3\u7a2e\u610f\u7fa9\u4e0b\uff0c\u767c\u751f\u7684\u6a5f\u7387\u3002<\/p>\n

    \u4f46\u5982\u524d\u6240\u8ff0\uff0c\u6211\u5011\u4e5f\u53ef\u5b8c\u5168\u4e0d\u7ba1\uff0c\u7a76\u7adf\u662f\u4e0d\u662f\u5728\u89c0\u6e2c\u4efb\u4f55\u96a8\u6a5f\u73fe\u8c61\uff0c\u4e5f\u5c31\u662f $$\\Omega$$ \u53ea\u8981\u662f\u4e00\u975e\u7a7a\u7684\u96c6\u5408\u5c31\u597d\u3002\u81f3\u65bc\u5c0d $$P(A)$$\uff0c\u5c31\u662f\u88ab\u7a31\u70ba\u4e8b\u4ef6 $$A$$ \u7684\u4e00\u500b\u51fd\u6578\u503c\uff0c \u6b64\u51fd\u6578\u503c\u5728\u6211\u5011\u7684\u8a0e\u8ad6\u88e1\uff0c\u53c8\u88ab\u7a31\u70ba\u4e8b\u4ef6$$A$$\u4e4b\u6a5f\u7387\u3002<\/p>\n

    \u4f46\u9019\u4e26\u7121\u95dc\u7dca\u8981\uff0c\u53ef\u4ee5\u4e0d\u5fc5\u806f\u60f3\u5230\u662f\u4e0d\u662f\u771f\u6709\u67d0\u4e00\u500b\u4e8b\u4ef6\u767c\u751f\u3002\u9019\u7a2e\u601d\u7dad\u4e26\u4e0d\u5947\u602a\u3002\u50cf\u662f\u559c\u6b61\u91d1\u5eb8\u6b66\u4fe0\u5c0f\u8aaa\u7684\u8b80\u8005\uff0c\u901a\u5e38\u4e26\u4e0d\u6703\u8ffd\u8457\u91d1\u5eb8\u554f\u6b77\u53f2\u4e0a\u662f\u5426\u771f\u6709\u90ed\u9756\u53ca\u9ec3\u84c9\u7b49\u3002\u4f46\u91d1\u5eb8\u5c0f\u8aaa\u4e2d\u7684\u5408\u7406\u6027\u537b\u662f\u8b80\u8005\u53ef\u4ee5\u8ffd\u7a76\u7684\u3002\u56e0\u6b64\uff0c\u4ee5\u516c\u7406\u5316\u7684\u7684\u65b9\u5f0f\u4f86\u5b9a\u7fa9\u6a5f\u7387\uff0c\u4e5f\u8981\u80fd\u81ea\u5713\u5176\u8aaa\uff0c\u6216\u8fd1\u4e00\u6b65\u8981\u6c42\uff0c\u8981\u5929\u8863\u7121\u7e2b\u3002<\/p>\n

    \u4f46\u6bd4\u5c0f\u8aaa\u7684\u865b\u69cb\u66f4\u5be6\u969b\u7684\u662f\uff0c\u6211\u5011\u96d6\u4ee5\u62bd\u8c61\u7684\u65b9\u5f0f\u5f15\u9032\u6a5f\u7387\uff0c\u4f46\u53ef\u5957\u9032\u4efb\u4f55\u5be6\u969b(\u6216\u865b\u64ec)\u7684\u96a8\u6a5f\u73fe\u8c61\u3002\u4e5f\u5c31\u662f\u4e00\u65e6\u6558\u660e\uff0c$$\\Omega$$ \u662f\u89c0\u6e2c\u90a3\u4e00\u96a8\u6a5f\u73fe\u8c61\u6240\u5f97\u4e4b\u5168\u90e8\u53ef\u80fd\u7684\u7d50\u679c\u4e4b\u96c6\u5408\uff0c\u5247 $$P(A)$$ \u4fbf\u70ba\u4e00\u7279\u5b9a\u4e8b\u4ef6 $$A$$ \u6703\u767c\u751f\u4e4b\u6a5f\u7387\u4e86\u3002<\/p>\n

    \u5f88\u591a\u6642\u5019\uff08\u5c24\u5176\u662f $$\\Omega$$ \u4e2d\u53ea\u6709\u6709\u9650\u591a\u500b\u5143\u7d20\uff09\uff0c\u5f80\u5f80\u5c07 $$\\mathcal{F}$$ \u53d6\u6210\u5305\u542b $$\\Omega$$ \u4e4b\u6240\u6709\u5b50\u96c6\u4e4b\u96c6\u5408\u3002\u9019\u6a23\u4e0d\u662f\u7c21\u55ae\u660e\u77ad\u55ce? \u70ba\u4ec0\u9ebc\u5728\u524d\u8ff0\u4e09\u57fa\u672c\u8981\u4ef6\u4e2d\uff0c\u6211\u5011\u537b\u5c07 $$\\mathcal{F}$$ \u53ea\u53d6\u6210 $$\\Omega$$ \u7684\u4e00\u4e9b\u5b50\u96c6\u6240\u5f62\u6210\u4e4b\u96c6\u5408?\u9019\u6a23\u7684\u4f5c\u6cd5\uff0c\u5176\u5be6\u662f\u6bd4\u8f03\u4e00\u822c\u3002<\/p>\n

    \u6709\u5982\u67d0\u73ed\u8981\u7d44\u4e00\u5566\u5566\u968a\u3002\u53ef\u4ee5\u5168\u73ed\u7686\u53c3\u8207\uff0c\u4e5f\u53ef\u4ee5\u53ea\u7531\u90e8\u5206\u540c\u5b78\u7d44\u6210\u3002\u4f46\u65e2\u7136\u662f\u5566\u5566\u968a\uff0c\u8981\u9054\u5230\u67d0\u4e00\u7a0b\u5ea6\u7684\u6f14\u51fa\u6548\u679c\uff0c\u4e26\u73a9\u51fa\u5920\u591a\u7684\u82b1\u6a23\uff0c\u4eba\u6578\u4fbf\u4e0d\u80fd\u904e\u5c11\uff0c\u751a\u81f3\u6709\u53ef\u80fd\u56e0\u591a\u627e\u4e86\u4e00\u500b\u5973\u751f\uff0c\u4fbf\u8981\u591a\u627e\u4e00\u4f4d\u7537\u751f\u8207\u5979\u914d\u3002$$\\mathcal{F}$$ \u4e5f\u662f\u4e00\u6a23\uff0c\u56e0\u53ea\u6709 $$\\mathcal{F}$$ \u4e2d\u7684\u5143\u7d20\uff0c\u624d\u7a31\u70ba\u4e8b\u4ef6\uff0c\u624d\u7d66\u5176\u6a5f\u7387\uff0c$$\\mathcal{F}$$ \u53ef\u4ee5\u5f88\u6709\u5f48\u6027\u7684\u627e\u4e00\u4e9b $$\\Omega$$ \u7684\u5b50\u96c6\u4f86\u7d44\u6210\uff0c\u4f46\u8981\u7d66\u5b9a\u4e00\u4e9b\u898f\u7bc4\uff0c\u4f7f $$\\mathcal{F}$$ \u5305\u542b\u5920\u591a\u8a72\u6709\u7684 $$\\Omega$$ \u4e4b\u5b50\u96c6\uff0c\u56e0\u6b64\u624d\u6709\u5920\u591a\u7684\u7406\u8ad6\u80fd\u767c\u5c55\u51fa\u4f86\u3002<\/p>\n

    \u90a3 $$\\mathcal{F}$$ \u8981\u6eff\u8db3\u90a3\u4e9b\u689d\u4ef6\u5462? \u5c0d\u65bc\u4e8c\u4e8b\u4ef6 $$A$$\uff0c$$B$$\uff08\u5373 $$A,B$$ \u7686\u5c6c\u65bc $$\\mathcal{F}$$\uff09\uff0c\u6709\u6642\u6211\u5011\u6703\u60f3\u77e5\u9053 $$A$$ \u6216 $$B$$ \u767c\u751f\uff08\u5373 $$A\\cup B$$ \u767c\u751f\uff09\u7684\u6a5f\u7387\u3002\u6709\u6642\u4e5f\u6703\u60f3\u77e5\u9053 $$A$$ \u8207 $$B$$ \u540c\u6642\u767c\u751f\uff08\u5373 $$A\\cap B$$ \u767c\u751f\uff09\u7684\u6a5f\u7387\u3002\u8b6c\u5982\u8aaa\u5c0d\u4e16\u754c\u76c3\u8db3\u7403\u8cfd\uff0c\u6709\u4eba\u652f\u6301\u5169\u652f\u7403\u968a\uff0c\u4ed6\u6703\u95dc\u5fc3\u6bcf\u4e00\u652f\u7403\u968a\u8d0f\u7403\u7684\u6a5f\u7387\uff08$$P(A)$$ \u53ca $$P(B)$$\uff09\uff0c\u4e5f\u6703\u95dc\u5fc3\u5169\u652f\u4e2d\u81f3\u5c11\u6709\u4e00\u652f\u8d0f\u7403\u7684\u6a5f\u7387\uff08$$P(A\\cup B)$$\uff09\uff0c\u4ee5\u53ca\u5169\u652f\u540c\u6642\u8d0f\u7403\u7684\u6a5f\u7387\uff08$$P(A\\cap B)$$\uff09\u3002\u5c0d\u4e8b\u4ef6 $$A$$ \u6709\u8208\u8da3\uff0c\u5176\u4e0d\u767c\u751f\uff08\u5373 $$A$$ \u7684\u9918\u96c6 $$A^c$$ \u767c\u751f\uff09\u7684\u6a5f\u7387\u4e5f\u5f88\u96e3\u4e0d\u60f3\u77e5\u9053\u3002\u6240\u4ee5 $$\\mathcal{F}$$ \u61c9\u8981\u6eff\u8db3<\/p>\n

      \n
    1. \u82e5 $$A\\in\\mathcal{F}$$\uff0c\u5247\u00a0$$A^c\\in\\mathcal{F}$$<\/li>\n
    2. \u82e5 $$A,B\\in\\mathcal{F}$$\uff0c\u5247\u00a0$$A\\cup B\\in\\mathcal{F}$$\uff0c\u4e14\u00a0$$A\\cap B\\in\\mathcal{F}$$<\/li>\n<\/ol>\n

      \u7531\u4e0a\u8ff0\u689d\u4ef6 (b)\uff0c\u7acb\u5373\u5f97<\/p>\n

      \u5c0d $$\\forall n\\ge 1$$\uff0c\u82e5 $$A_1,A_2,\\cdots,A_n\\in\\mathcal{F}$$\uff0c<\/p>\n

      \u5247 $$\\bigcup_{i=1}^{n}A_i=A_1\\cup A_2\\cup \\cdots\\cup A_n$$\uff0c\u53ca\u00a0$$\\bigcap_{i=1}^{n}A_i=A_1\\cap A_2\\cap \\cdots\\cap A_n$$ \u7686\u5c6c\u65bc $$\\mathcal{F}$$\u3002<\/p>\n

      \u9019\u53ea\u8981\u7531 $$A_1\\cup A_2\\cup A_3=(A_1\\cup A_2)\\cup A_3$$\uff0c$$A_1\\cap A_2\\cap A_3=(A_1\\cap A_2)\\cap A_3$$ \u4fbf\u53ef\u770b\u51fa\u3002<\/p>\n

      \u82e5 $$\\Omega$$ \u53ea\u662f\u4e00\u6709\u9650\u7684\u96c6\u5408\uff0c\u5c0d $$\\mathcal{F}$$ \u8981\u6c42\u6eff\u8db3\u689d\u4ef6(a)\u53ca(b)\u4fbf\u5920\u4e86\u3002\u4f46\u82e5 $$\\Omega$$ \u70ba\u4e00\u7121\u9650\u96c6\u5408\uff0c\u53ea\u5047\u8a2d\u689d\u4ef6(a)\u53ca(b)\uff0c\u537b\u7121\u6cd5\u5c0e\u51fa\u5920\u591a\u7684\u6578\u5b78\u7406\u8ad6\u3002<\/p>\n

      \u6578\u5b78\u4e2d\u5e38\u4e0d\u5f97\u4e0d\u8a0e\u8ad6\u6975\u9650\uff0c\u5982\u679c\u6709\u4e00\u6578\u5217\u7684\u4e8b\u4ef6$$A_1,A_2,\\cdots$$ \u7686\u5c6c\u65bc $$\\mathcal{F}$$\uff0c<\/p>\n

      \u5247\u5c0d\u65bc $$\\bigcup^\\infty_{i=1}A_i=A_1\\cup A_2\\cup\\cdots$$\uff0c$$\\bigcap^\\infty_{i=1}A_i=A_1\\cap A_2\\cap\\cdots$$\uff0c
      \n\u6211\u5011\u6703\u5e0c\u671b\u4e8c\u8005\u7686\u5c6c\u65bc $$\\mathcal{F}$$\u3002<\/p>\n

      \u5be6\u969b\u4e0a\uff0c\u7531\u65bc\u6709\u72c4\u83ab\u6839\u6cd5\u5247(De Morgan’s laws)\uff1a\u5c0d\u4efb\u4e00\u6578\u5217\u4e4b\u96c6\u5408 $$B_1,B_2,\\cdots$$<\/p>\n

      $$(1)~~~(\\bigcup^\\infty_{i=1}B_i)^c=\\bigcap^\\infty_{i=1}B_i^c$$<\/p>\n

      \u4e14<\/p>\n

      $$(2)~~~(\\bigcap^\\infty_{i=1}B_i)^c=\\bigcup^\\infty_{i=1}B_i^c$$<\/p>\n

      \u56e0\u6b64\u53ea\u9700\u8981\u6c42 $$\\cup^\\infty_{i=1}A_i$$ \u8207\u00a0$$\\cap^\\infty_{i=1}A_i$$\uff0c\u5169\u8005\u4e4b\u4e00\u5c6c\u65bc\u00a0$$\\mathcal{F}$$ \u5373\u53ef\u3002<\/p>\n

      \u9019\u662f\u56e0\u5df2\u6709\u300c\u82e5 $$A\\in\\mathcal{F}$$\uff0c\u5247 $$A^c\\mathcal{F}$$\u300d\uff0c\u6545<\/p>\n

      $$\\bigcap^\\infty_{i=1}A_i=(\\bigcup^\\infty_{i=1}A_i^c)^c$$\uff0c$$\\bigcup^\\infty_{i=1}A_i=(\\bigcap^\\infty_{i=1}A_i^c)^c$$<\/p>\n

      \u6b64\u8655\u5206\u5225\u7528\u5230 $$(1)$$ \u53ca $$(2)$$ \u5f0f\uff0c\u4ee5\u53ca $$(A^c)^c=A$$ \u7b49\u6027\u8cea\u3002\u7e3d\u7d50\u5982\u4e0b: \u6211\u5011\u8981\u6c42 $$\\mathcal{F}$$ \u9808\u6eff\u8db3<\/p>\n

        \n
      1. \u82e5 $$A\\in\\mathcal{F}$$\uff0c\u5247 $$A^c\\mathcal{F}$$<\/li>\n
      2. \u82e5 $$A_1,A_,2,\\cdots\\in\\mathcal{F}$$\uff0c\u5247 $$\\cup^\\infty_{i=1}A_i\\in\\mathcal{F}$$<\/li>\n<\/ol>\n

        \u7576\u7136\u6211\u5011\u5df2\u6307\u51fa\uff0c \u6eff\u8db3(i)\u53ca(ii)\u7684 $$\\mathcal{F}$$\uff0c\u4fbf\u4ea6\u6eff\u8db3<\/p>\n

          \n
        1. \u82e5 $$A_1,A_,2,\\cdots\\in\\mathcal{F}$$\uff0c\u5247 $$\\cap^\\infty_{i=1}A_i\\in\\mathcal{F}$$<\/li>\n<\/ol>\n

          \u5728\u6b64\u6ce2\u745e\u723e(Emile Borel\uff0c1871-1956)\uff0c\u70ba\u8457\u540d\u7684\u6cd5\u570b\u6578\u5b78\u5bb6\uff0c \u4e5f\u662f\u73fe\u4ee3\u6a5f\u7387\u8ad6\u7684\u5275\u59cb\u8005\u4e4b\u4e00\u3002<\/p>\n

          \u5b9a\u7fa91.<\/span><\/strong> \u4e00\u975e\u7a7a\u96c6\u5408 $$K$$ \u4e4b\u4e00\u4e9b(\u81f3\u5c11\u4e00\u500b)\u5b50\u96c6\u6240\u5f62\u6210\u4e4b\u96c6\u5408 $$\\mathcal{L}$$\uff0c\u7a31\u70ba\u4e00 $$\\sigma$$-\u9ad4\uff08$$\\sigma$$-field\uff0c\u53c8\u7a31 $$\\sigma$$-algebra\uff0c\u6216 Borel field\uff09\uff0c\u82e5\u6eff\u8db3<\/p>\n

            \n
          1. \u82e5 $$B\\in\\mathcal{L}$$\uff0c\u5247 $$B^c\\in\\mathcal{L}$$<\/li>\n
          2. \u82e5 $$B_1,B_2,\\cdots\\in\\mathcal{L}$$\uff0c\u5247\u00a0\u5247 $$\\cup^\\infty_{i=1}B_i\\in\\mathcal{L}$$<\/li>\n<\/ol>\n

            \u6240\u4ee5\u6211\u5011\u5c31\u662f\u8981\u6c42 $$\\mathcal{F}$$ \u70ba\u4e00 $$\\sigma$$-\u9ad4\u3002<\/p>\n

            \u5c0d\u65bc $$\\sigma$$-\u9ad4 $$\\mathcal{F}$$\uff0c\u7acb\u5373\u4fbf\u6709 $$\\varnothing$$ \u53ca $$\\Omega$$ \u7686\u5c6c\u65bc $$\\mathcal{F}$$\uff08\u56e0 $$A\\cup A^c=\\Omega$$\uff0c\u800c $$\\Omega^c=\\varnothing$$\uff09\u3002<\/p>\n

            \u5373 $$\\{\\varnothing,\\Omega\\}$$ \u70ba\u7531 $$\\Omega$$ \u6240\u7522\u751f\u4e4b\u6700\u5c0f\u7684 $$\\sigma$$-\u9ad4\u3002\u6240\u8b02\u6700\u5c0f\uff0c\u662f\u6307\u82e5\u6709\u5176\u4ed6 $$\\sigma$$-\u9ad4\uff0c\u4e00\u5b9a\u5305\u542b\u9019\u4e00\u500b\u3002<\/p>\n

            \u800c $$\\Omega$$ \u6240\u6709\u5b50\u96c6\u4e4b\u96c6\u5408\uff0c\u53ef\u8996\u70ba\u6700\u5927\u7684 $$\\sigma$$-\u9ad4\u3002<\/p>\n

            \u7531 $$\\mathcal{F}$$ \u9808\u6eff\u8db3\u7684\u689d\u4ef6(ii)\uff0c\u7acb\u5373\u5c0e\u81f4\u524d\u8ff0\u689d\u4ef6(b)\u6210\u7acb\uff08\u53ea\u8981\u5c07 $$A_{n+1},A_{n+2},\\cdots$$\uff0c\u7686\u53d6\u6210 $$\\varnothing$$\uff0c<\/p>\n

            \u5247 $$\\bigcup^\\infty_{i=1}A_i=\\bigcup^n_{i=1}A_i$$\uff09;\u4e14\u82e5 $$\\Omega$$ \u70ba\u4e00\u6709\u9650\u96c6\u5408\uff0c\u5247\u4e8c\u689d\u4ef6\u7b49\u50f9\u3002<\/p>\n

            \u73fe\u5728\u6211\u5011\u53ef\u4ee5\u5b9a\u7fa9\u6a5f\u7387\u51fd\u6578(probability function)\u4e86\u3002<\/p>\n

            \u5b9a\u7fa92.<\/span><\/strong> \u8a2d $$\\Omega$$ \u70ba\u4e00\u6a23\u672c\u7a7a\u9593\uff0c$$\\mathcal{F}$$ \u70ba\u00a0$$\\Omega$$ \u4e4b\u4e00\u4e9b\u5b50\u96c6\u6240\u5f62\u6210\u4e4b\u4e00 $$\\sigma$$-\u9ad4\u3002\u5247\u4ee5 $$\\mathcal{F}$$ \u70ba\u5b9a\u7fa9\u57df\uff0c\u6eff\u8db3\u4e0b\u8ff0\u689d\u4ef6\u7684\u51fd\u6578 $$P$$\uff0c\u4fbf\u7a31\u70ba\u4e00\u6a5f\u7387\u51fd\u6578\uff1a<\/p>\n

              \n
            1. $$\\forall A\\in\\mathcal{F},~P(A)\\ge 0$$<\/li>\n
            2. $$P(\\Omega)=1$$<\/li>\n
            3. \u82e5 $$A_1,A_2,\\cdots\\in\\mathcal{F}$$\uff0c\u4e14 $$A_i\\cap A_j=\\varnothing,~\\forall i\\ne j$$\uff0c\u5247<\/li>\n<\/ol>\n

              $$(3)~~~P(\\bigcup^\\infty_{i=1}A_i)=\\sum^\\infty_{i=1}P(A_i)$$<\/p>\n

              \u5c0d\u4e00\u6a23\u672c\u7a7a\u9593 $$\\Omega$$\uff0c \u6211\u5011\u5f15\u9032\u4e86 $$\\sigma$$-\u9ad4 $$\\mathcal{F}$$\uff0c\u53ca\u6a5f\u7387\u51fd\u6578 $$P$$\uff0c<\/p>\n

              $$(\\Omega,\\mathcal{F},P)$$ \u4fbf\u69cb\u6210\u4e00\u6a5f\u7387\u7a7a\u9593(probability space)\u3002<\/p>\n

              \u5b9a\u7fa92\u4e2d\u7684 $$(3)$$ \u689d\u4ef6\uff0c\u901a\u5e38\u7a31\u70ba\u6a5f\u7387\u7684\u516c\u7406\u6216\u67ef\u83ab\u679c\u6d1b\u592b\u516c\u7406(Kolmogorov axioms)\u3002<\/p>\n

              \u5c0d\u540c\u4e00\u6a23\u672c\u7a7a\u9593\uff0c\u53ef\u4ee5\u6709\u4e0d\u540c\u7684 $$\\sigma$$-\u9ad4\u3002\u5373\u4f7f $$\\sigma$$-\u9ad4\u76f8\u540c\uff0c\u4e5f\u53ef\u6709\u4e0d\u540c\u7684\u6a5f\u7387\u51fd\u6578\u3002<\/p>\n

              \u5c0d $$\\forall A\\in \\mathcal{F}$$\uff0c$$A$$ \u7a31\u70ba\u4e00\u4e8b\u4ef6\uff0c$$P(A)$$ \u7a31\u70ba $$A$$ \u4e4b\u6a5f\u7387\u3002<\/p>\n

              \u81f3\u65bc\u82e5 $$B\\subset\\Omega$$\uff0c\u800c $$B\\not\\in\\mathcal{F}$$\uff0c\u5247 $$B$$ \u4e0d\u70ba\u4e8b\u4ef6\uff0c\u4e0d\u7528\u77e5\u9053(\u4e8b\u5be6\u4e0a\u4e5f\u6c92\u7d66)\u5176\u6a5f\u7387\u3002<\/p>\n

              \u6a5f\u7387\u7a7a\u9593\u662f\u6a5f\u7387\u8ad6\u7684\u57fa\u790e\u3002\u82e5\u53ea\u662f\u8655\u7406\u4e00\u4e9b\u7c21\u55ae\u7684\u60c5\u6cc1\u4e0b\u7684\u6a5f\u7387\uff0c\u5247\u6709\u6642\u4e0d\u7528\u592a\u5728\u4e4e\u6240\u6d89\u53ca\u7684\u6a5f\u7387\u7a7a\u9593\u7a76\u7adf\u662f\u4ec0\u9ebc\u3002\u4f46\u5728\u8655\u7406\u8f03\u7d30\u81a9\u7684\u554f\u984c\u6642\uff0c\u6709\u6642\u5c31\u5f97\u5f04\u6e05\u695a\u6a5f\u7387\u7a7a\u9593\u7a76\u7adf\u70ba\u4f55\u3002\u9019\u5f77\u5f7f\u5c0d\u67d0\u4e00\u500b\u4eba\uff0c\u6709\u6642\u6211\u5011\u53ea\u9700\u8981\u77e5\u9053\u4ed6\u7684\u540d\u5b57\uff0c\u6709\u6642\u5f97\u9032\u4e00\u6b65\u4e86\u89e3\u9019\u500b\u4eba\u7684\u300c\u51fa\u8eab\u300d\uff0c\u5982\u5b78\u7d93\u6b77\u7b49\u3002<\/p>\n

              \u70ba\u4e86\u7c21\u4fbf\uff0c\u7279\u5225\u662f $$\\Omega$$ \u4e2d\u53ea\u6709\u53ef\u6578\u7684(countable)\u591a\u4e4b\u5143\u7d20\u6642\uff0c\u5e38\u5c07 $$\\mathcal{F}$$ \u53d6\u6210 $$\\Omega$$ \u4e4b\u6240\u6709\u5b50\u96c6\u4e4b\u96c6\u5408\u3002\u5373\u4f7f $$\\Omega$$ \u662f\u4e00\u7121\u9650\u5340\u9593\uff0c\u8b6c\u5982\u8aaa\u5be6\u6578\u7684\u96c6\u5408\uff0c$$\\mathcal{F}$$ \u4e5f\u6709\u4e00\u5e38\u898b\u7684\u53d6\u6cd5\u3002 \u6211\u5011\u7a0d\u5f8c\u518d\u4ecb\u7d39\u3002\u5728\u5f88\u591a\u5be6\u4f8b\u4e2d\uff0c\u6703\u8aaa\u660e $$\\Omega$$ \u53ca $$P$$ \u5404\u662f\u4ec0\u9ebc\uff0c\u537b\u672a\u898b\u5230 $$\\mathcal{F}$$\u3002\u6b64\u5373\u8868 $$\\mathcal{F}$$ \u662f\u53d6\u6210\u5e38\u898b\u7684 $$\\sigma$$-\u9ad4\uff0c\u56e0\u6b64\u4e0d\u7528\u7279\u5225\u63d0\u5b83\u3002<\/p>\n

              \u8a3b1\uff1a\u4e00\u96c6\u5408\u7a31\u70ba\u53ef\u6578\u7684\u82e5\u5176\u70ba\u6709\u9650\uff0c\u6216\u7121\u9650\u4f46\u8207\u6b63\u6574\u6578\u96c6\u5408\u6709\u4e00\u5c0d\u4e00\u4e14\u6620\u6210\u7684\u5c0d\u61c9(\u5373\u53ef\u8868\u70ba$$\\{a_1,a_2,\\cdots\\}$$)\u3002<\/p>\n

              \u6a5f\u7387\u51fd\u6578\u6709\u4e00\u4e9b\u57fa\u672c\u7684\u6027\u8cea\uff0c \u6211\u5011\u5217\u8209\u4e00\u4e9b\u5982\u4e0b\u3002<\/p>\n

              \u5b9a\u74061.<\/span> \u8a2d $$(\\Omega,\\mathcal{F},P)$$ \u70ba\u4e00\u6a5f\u7387\u7a7a\u9593\u3002\u5247<\/p>\n

                \n
              1. $$P(\\varnothing)=0$$<\/li>\n
              2. $$P(A)\\le 1,~\\forall A\\in\\mathcal{F}$$<\/li>\n
              3. $$P(A^c)=1-P(A),~\\forall A\\in\\mathcal{F}$$<\/li>\n
              4. $$P(A\\cup B)=P(A)+P(B)-P(A\\cap B),~\\forall A,B\\in\\mathcal{F}$$<\/li>\n
              5. $$P(A\\cap B)\\ge P(A)+P(B)-1,~\\forall A,B\\in\\mathcal{F}$$<\/li>\n
              6. \u8a2d $$C_1,C_2,\\cdots \\in\\mathcal{F},~C_i\\cap C_j=\\varnothing,~\\forall i\\ne j$$\uff0c\u4e14 $$\\cup^{\\infty}_{i=1}C_i=\\Omega$$\uff0c\u5247<\/li>\n<\/ol>\n

                $$(4)~~~P(A)=\\sum_{i=1}^{\\infty}P(A\\cap C_i),~\\forall A\\in\\mathcal{F}$$<\/p>\n

                  \n
                1. \u5c0d\u4efb\u610f $$A_1,A_2,\\cdots \\in \\mathcal{F}$$<\/li>\n<\/ol>\n

                  $$(5)~~~P(\\bigcup_{i=1}^{\\infty}A_i)\\le \\sum_{i=1}^{\\infty}P(A_i)$$<\/p>\n

                  \u5728\u5b9a\u74061\u4e2d\u4e4b(vi)\uff0c\u6eff\u8db3\u6240\u5217\u689d\u4ef6\u4e4b $$C_1,C_2,\\cdots$$\uff0c\u7a31\u70ba $$\\Omega$$ \u4e4b\u4e00\u5206\u5272(partition)\u3002<\/p>\n

                  \u4e8b\u4ef6 $$A_1,A_2,\\cdots$$(\u6709\u9650\u500b\u4ea6\u53ef)\uff0c\u82e5\u6eff\u8db3 $$A_i\\cap A_j=\\varnothing,~\\forall i\\neq j$$\uff0c\u4fbf\u7a31\u70ba\u6bcf\u5c0d\u4e92\u65a5(pairwise disjoint)\u3002\u81f3\u65bc\u82e5 $$A,B\\in\\mathcal{F}$$\uff0c\u4e14 $$A\\cap B=\\varnothing$$\uff0c\u4fbf\u7a31 $$A$$ \u8207 $$B$$ \u4e92\u65a5(disjoint)\u3002<\/p>\n

                  $$(5)$$ \u5f0f\u5373\u70ba\u6ce2\u723e\u4e0d\u7b49\u5f0f(Boole’s inequality)\u3002\u6ce2\u723e(George Boole\uff0c1815-1864)\u70ba\u82f1\u570b\u6578\u5b78\u5bb6\u3002<\/p>\n

                  \u6a5f\u7387\u51fd\u6578\u5c1a\u6709\u8a31\u591a\u5176\u4ed6\u7684\u6027\u8cea\uff0c \u6b64\u8655\u66ab\u4e14\u5217\u4e0b\u4e0d\u8868\u3002\u4e0d\u904e\u4e00\u500b\u770b\u8d77\u4f86\u8981\u6c42\u4e26\u4e0d\u592a\u591a\u7684\u516c\u7406\u5316\u7684\u7cfb\u7d71\uff0c\u5c31\u5f15\u767c\u51fa\u8a31\u591a\u6709\u8da3\u4e14\u503c\u5f97\u63a2\u8a0e\u7684\u7406\u8ad6\uff0c\u4e14\u61c9\u7528\u5ee3\u6cdb\u3002\u81ea\u67ef\u83ab\u679c\u6d1b\u592b\u4e4b\u5f8c\uff0c\u6a5f\u7387\u8ad6\u4fbf\u9010\u6f38\u70ba\u6578\u5b78\u5bb6\u6240\u63a5\u53d7\u3002\u6642\u81f3\u4eca\u65e5\uff0c\u6a5f\u7387\u8ad6\u5df2\u6210\u70ba\u6578\u5b78\u4e2d\u4e00\u91cd\u8981\u7684\u9818\u57df\uff0c\u53ef\u8207\u90a3\u4e9b\u50b3\u7d71\u7684\u4ee3\u6578\u3001\u5206\u6790\u53ca\u5e7e\u4f55\u7b49\u9818\u57df\u5206\u5ead\u6297\u79ae\u4e86\u3002<\/p>\n

                  \u9023\u7d50\uff1a\u6a5f\u7387\u7a7a\u9593\uff084\uff09\u6a5f\u7387\u7a7a\u9593\u4e4b\u4f8b<\/a><\/p>\n

                  \u53c3\u8003\u8cc7\u6599\uff1a<\/p>\n

                    \n
                  1. \u9ec3\u6587\u748b (2003). \u6578\u7406\u7d71\u8a08\u3002\u83ef\u6cf0\u6587\u5316\u4e8b\u696d\u80a1\u4efd\u6709\u9650\u516c\u53f8\uff0c\u53f0\u5317\u3002<\/li>\n<\/ol>\n<\/div>","protected":false},"excerpt":{"rendered":"

                    \u6a5f\u7387\u7a7a\u9593\uff083\uff09\u6a5f\u7387\u7a7a\u9593\uff08Probability space-3. Probability space\uff09 \u570b\u7acb\u9ad8…<\/p>\n","protected":false},"author":50,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[111,229],"tags":[4372,4369,3510,4370,3509,2384,3911,4371],"class_list":["post-39709","post","type-post","status-publish","format-standard","hentry","category-mathematics00","category-math06","tag-kolmogorov","tag-4369","tag-3510","tag-4370","tag-3509","tag-2384","tag-3911","tag-booles-inequality","loop-entry","cat-111","cat-229","no-thumbnail"],"views":7914,"_links":{"self":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/39709","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/users\/50"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/comments?post=39709"}],"version-history":[{"count":1,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/39709\/revisions"}],"predecessor-version":[{"id":87817,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/39709\/revisions\/87817"}],"wp:attachment":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/media?parent=39709"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/categories?post=39709"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/tags?post=39709"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}