{"id":13070,"date":"2024-05-14T02:42:22","date_gmt":"2024-05-14T02:42:22","guid":{"rendered":"https:\/\/science-hub.click\/%E3%83%AB%E3%83%99%E3%83%BC%E3%82%B0%E9%83%A8%E6%97%8F-%E5%AE%9A%E7%BE%A9\/"},"modified":"2024-05-14T02:42:22","modified_gmt":"2024-05-14T02:42:22","slug":"%E3%83%AB%E3%83%99%E3%83%BC%E3%82%B0%E9%83%A8%E6%97%8F-%E5%AE%9A%E7%BE%A9","status":"publish","type":"post","link":"https:\/\/science-hub.click\/?p=13070","title":{"rendered":"\u30eb\u30d9\u30fc\u30b0\u90e8\u65cf – \u5b9a\u7fa9"},"content":{"rendered":"

\u5c0e\u5165<\/h2>

\u30eb\u30d9\u30fc\u30b0\u53ef\u6e2c\u96c6\u5408<\/b>(\u3057\u3070\u3057\u3070\u53ef\u6e2c<\/b>\u3068\u7701\u7565\u3055\u308c\u308b) \u306f\u7a7a\u9593\u306e\u4e00\u90e8\u3067\u3059

$$ {\\R^n} $$<\/div>\u305d\u306e\u30eb\u30d9\u30fc\u30b0\u6e2c\u5ea6\u306f\u5b9a\u7fa9\u3067\u304d\u3001\u3053\u306e\u6982\u5ff5\u306f\u4efb\u610f\u306e\u5fae\u5206\u53ef\u80fd\u306a\u591a\u69d8\u4f53M<\/i><\/span>\u306b\u62e1\u5f35\u3067\u304d\u307e\u3059\u3002\u30eb\u30d9\u30fc\u30b0\u65cf\u3092<\/b>M<\/i><\/span>\u306e\u30eb\u30d9\u30fc\u30b0\u53ef\u6e2c\u90e8\u5206\u306e\u96c6\u5408<\/a><\/span>\u3068\u547c\u3073\u307e\u3059\u3002<\/p>
\n\"\u30eb\u30d9\u30fc\u30b0\u90e8\u65cf<\/figure>

\u610f\u5473<\/h2>

\u30eb\u30d9\u30fc\u30b0\u6e2c\u5ea6\u306e<\/a><\/span>\u8a18\u4e8b\u3067\u8aac\u660e\u3055\u308c\u3066\u3044\u308b\u3088\u3046\u306b\u3001\u3053\u306e\u6e2c\u5ea6\u306f

$$ {\\R^n} $$<\/div>\u306e\u90e8\u5206\u306e\u03c3\u4ee3\u6570\u3067\u5b9a\u7fa9\u3055\u308c\u308b
$$ {\\R^n} $$<\/div> \u3001\u30dc\u30ec\u30ea\u30a2\u65cf\u306b\u3088\u3063\u3066\u88dc\u8db3\u3055\u308c\u307e\u3057\u305f\u3002\u3053\u306e\u90e8\u65cf\u306f\u30eb\u30d9\u30fc\u30b0\u65cf<\/b>\u3068\u547c\u3070\u308c\u3001\u305d\u308c\u3092\u69cb\u6210\u3059\u308b\u96c6\u5408\u306f\u30eb\u30d9\u30fc\u30b0\u3067\u6e2c\u5b9a\u53ef\u80fd\u306a\u90e8\u5206<\/b>\u3067\u3059\u3002
$$ {\\R^n} $$<\/div> \u3002<\/p>

\u30eb\u30d9\u30fc\u30b0\u65cf\u306e\u67a2\u6a5f\u537f<\/h2>

\u63d0\u6848<\/strong>\u2014<\/span>\u30eb\u30d9\u30fc\u30b0\u65cf\u306e\u67a2\u6a5f\u537f

$$ {\\R^n} $$<\/div>\u306e\u3059\u3079\u3066\u306e\u90e8\u5206\u306e\u3053\u3068\u3067\u3059
$$ {\\R} $$<\/div> \u3002<\/p><\/div>

\u8a3c\u62e0<\/b>\uff1a<\/p>

\u306e\u305f\u3081\u306b

$$ {n\\geq2} $$<\/div>\u7c21\u5358\u3067\u3059\uff1a\u5168\u4f53
$$ {\\R^{n-1}\\times\\{0\\}} $$<\/div>\u306e\u30dc\u30ec\u30eb\u3067\u3059
$$ {\\R^n} $$<\/div>\u30bc\u30ed\u6e2c\u5b9a\u306e\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u305d\u306e\u3059\u3079\u3066\u306e\u90e8\u5206\u306f\u7121\u8996\u3067\u304d\u308b\u305f\u3081\u3001\u30eb\u30d9\u30fc\u30b0\u6e2c\u5b9a\u53ef\u80fd\u3067\u3059\u3002<\/p>

n<\/i> = 1<\/span>\u306e\u5834\u5408\u306f\u3001\u3082\u3046\u5c11\u3057\u5206\u304b\u308a\u306b\u304f\u3044\u4f8b\u3092\u63a2\u3059\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002\u30ab\u30f3\u30c8\u30fc\u30eb\u306e\u4e09\u9032\u96c6\u5408\u304c<\/span>\u7b54\u3048\u3092\u63d0\u4f9b\u3057\u307e\u3059\u3002\u305d\u308c\u306f\u30b3\u30f3\u30d1\u30af\u30c8\u306a\u96c6\u5408\u3067\u3042\u308a\u3001\u3057\u305f\u304c\u3063\u3066\u30dc\u30ec\u30ea\u30a2\u30f3\u3067\u3042\u308a\u3001\u30bc\u30ed\u5c0f\u7bc0\u3067\u3042\u308a<\/span>\u306a\u304c\u3089\u3001

$$ {\\R} $$<\/div> \u3002\u3057\u305f\u304c\u3063\u3066\u3001\u305d\u306e\u90e8\u5206\u306f\u30eb\u30d9\u30fc\u30b0\u53ef\u6e2c\u3067\u3042\u308a\u3001\u305d\u306e\u5168\u4f53\u306e\u57fa\u6570\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002
$$ {2^\\mathfrak c} $$<\/div> \u3001 \u307e\u305f\u306f
$$ {\\mathfrak c} $$<\/div> \uff5e\u306e\u67a2\u6a5f\u537f\u3092\u6307\u540d\u3059\u308b
$$ {\\R} $$<\/div> \uff08\u300c\u7d99\u7d9a\u306e\u529b\u300d\uff09\u3002<\/p>

QED<\/span><\/p>

\n\"\u30eb\u30d9\u30fc\u30b0\u90e8\u65cf<\/figure>

n \u6b21\u5143\u7a7a\u9593\u306b\u304a\u3051\u308b\u6e2c\u5b9a\u53ef\u80fd\u306a\u3082\u306e\u306e\u7279\u6027\u8a55\u4fa1<\/h2>

\u30dc\u30ec\u30eb\u65cf\u5b8c\u6210\u3068\u3044\u3046<\/a><\/span>\u89b3\u70b9<\/a><\/span>\u304b\u3089<\/span><\/h3>

\u6e2c\u5b9a\u53ef\u80fd\u306a\u90e8\u5206\u306f\u3001

$$ {\\R^n} $$<\/div>\u90e8\u5206 A \u306f\u6b21\u306e\u5f62\u5f0f\u3067\u8a18\u8ff0\u3067\u304d\u307e\u3059\u3002 <\/p>
$$ {A=B\\cup N} $$<\/div> \u3001 B<\/i><\/span>\u30dc\u30ec\u30ea\u30a2\u30f3\u3068N<\/i><\/span>\u306f\u7121\u8996\u3067\u304d\u307e\u3059 (\u30dc\u30ec\u30eb\u30fb\u30eb\u30d9\u30fc\u30b0\u6e2c\u5ea6\u306e\u5834\u5408)\u3002<\/center>

\u6b21\u306e\u5909\u5f62\u4f8b\u304c\u5f79\u7acb\u3064\u5834\u5408\u304c\u3042\u308a\u307e\u3059\u3002 A \u306f\u3001\u6b21\u306e\u5f62\u5f0f\u3067\u8a18\u8ff0\u3067\u304d\u308b\u5834\u5408\u306b\u306e\u307f\u6e2c\u5b9a\u53ef\u80fd\u3067\u3059\u3002 <\/p>

$$ {A=B\\,\\Delta\\, N} $$<\/div> \u3001 B \u306f<\/i><\/span>\u30dc\u30ec\u30ea\u30a2\u30f3\u3001 N \u306f<\/i><\/span>\u7121\u8996\u3067\u304d\u307e\u3059 ( \u0394 \u306f<\/span>\u5bfe\u79f0\u7684\u306a\u5dee\u3092\u8868\u3057\u307e\u3059)\u3002<\/center>

\u5916\u90e8\u6e2c\u5b9a\u306e\u89b3\u70b9\u304b\u3089<\/span><\/h3>

\u3053\u306e\u30bb\u30af\u30b7\u30e7\u30f3\u3067\u306f\u3001\u6b21\u306e\u70b9\u306b\u6ce8\u610f\u3057\u307e\u3059\u3002

$$ {\\mathcal{S}} $$<\/div> \u300c\u8217\u88c5\u300d\u306e\u96c6\u5408\u3001\u3064\u307e\u308a\u6709\u754c\u533a\u9593\u306e\u30c7\u30ab\u30eb\u30c8\u7a4d\u3001\u3064\u307e\u308a\u5f62\u5f0f\u306e\u96c6\u5408
$$ {E=I_1 \\times I_2 \\times \\cdots \\times I_n} $$<\/div> \u3001\u3053\u3053\u3067\u3001 I<\/i> i \u306f<\/i><\/sub><\/span>\u9593\u9694\u3092\u6307\u5b9a\u3057\u307e\u3059\u3002
$$ {\\R} $$<\/div>\u30af\u30ed\u30fc\u30ba\u3001\u30aa\u30fc\u30d7\u30f3\u3001\u307e\u305f\u306f\u30bb\u30df\u30aa\u30fc\u30d7\u30f3\u306e\u3044\u305a\u308c\u304b\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066\u304f\u3060\u3055\u3044\u3002
$$ {\\mathrm{Vol}\\,(E)} $$<\/div>\u305d\u306e\u3088\u3046\u306a\u30d6\u30ed\u30c3\u30af\u306e\u4f53\u7a4d\uff08\u8fba\u306e\u9577\u3055\u306e\u7a4d\u3068\u3044\u3046\u610f\u5473<\/a><\/span>\u3067\uff09\u3002<\/p>

\u306e\u305f\u3081\u306b

$$ {A\\subset \\R^n} $$<\/div> \u3001\u5916\u90e8\u6e2c\u5b9a<\/i>
$$ {\\lambda_n^*} $$<\/div> A<\/i><\/span>\u306e \u306f\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002 <\/p>
$$ {\\lambda_n^*(A)=\\mathrm{inf}\\{\\sum_{k=1}^{+\\infty}\\mathrm{Vol}\\,(E_k)\\,\\mid\\,E_k\\in\\mathcal{S},\\,A\\subset\\bigcup_{k=1}^{+\\infty}E_k\\}.} $$<\/div><\/center>

\u5b9a\u7406<\/span><\/strong>\u2014<\/span>\u3057\u307e\u3057\u3087\u3046

$$ {A\\subset \\R^n} $$<\/div> \u3002\u96c6\u5408A \u306f<\/i><\/span>\u3001\u6b21\u306e\u5834\u5408\u306b\u306e\u307f\u30eb\u30d9\u30fc\u30b0\u53ef\u6e2c\u3067\u3059\u3002<\/p>
\u3059\u3079\u3066\u306e<\/a><\/span>\u305f\u3081\u306b
$$ {X\\subset \\R^n} $$<\/div> \u3001
$$ {\\lambda_n^*(X\\cap A)+\\lambda_n^*(X\\setminus A)=\\lambda_n^*(X)} $$<\/div> \u3002<\/center><\/div>

\u3053\u306e\u7279\u5fb4\u4ed8\u3051\u306f\u30ab\u30e9\u30c6\u30aa\u30c9\u30ea\u306b\u3088\u308b\u3082\u306e\u3067\u3001\u30eb\u30d9\u30fc\u30b0\u306b\u3088\u308b\u5143\u306e\u7279\u5fb4\u4ed8\u3051\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3059\u3002<\/p>

\u5b9a\u7406<\/strong>\u2014<\/span>\u3057\u307e\u3057\u3087\u3046

$$ {A\\subset \\R^n} $$<\/div>\u6709\u754c\u3067\u3042\u308a\u3001 E \u306f<\/i><\/span>A \u3092<\/i><\/span>\u542b\u3080\u30d6\u30ed\u30c3\u30af\u3067\u3059\u3002\u96c6\u5408A \u306f<\/i><\/span>\u3001\u6b21\u306e\u5834\u5408\u306b\u306e\u307f\u30eb\u30d9\u30fc\u30b0\u53ef\u6e2c\u3067\u3059\u3002 <\/p>
$$ {\\lambda_n^*(A)+\\lambda_n^*(E\\setminus A)=\\mathrm{Vol}\\,(E)} $$<\/div> \u3002<\/center><\/div>

\u672c\u7269\u3067\u3042\u308b\u3053\u3068\u304c\u7c21\u5358\u306b\u5206\u304b\u308a\u307e\u3059

$$ {\\displaystyle\\lambda_{n*}(A)} $$<\/div>\u306b\u3088\u3063\u3066\u5b9a\u7fa9\u3055\u308c\u308b
$$ {\\lambda_{n*}(A)=\\mathrm{Vol}\\,(E)-\\lambda_n^*(E\\setminus A)} $$<\/div> A \u3092<\/i><\/span>\u30dc\u30c3\u30af\u30b9\u5316\u3059\u308b\u305f\u3081\u306b\u4f7f\u7528\u3055\u308c\u308b\u30bf\u30a4\u30eb\u3068\u306f\u72ec\u7acb\u3057\u3066\u3044\u307e\u3059\u3002\u3053\u308c\u3092\u5b9f\u969b\u306eA<\/i><\/span>\u306e\u300c\u5185\u90e8\u30e1\u30b8\u30e3\u30fc\u300d\u3068\u547c\u3073\u307e\u3059\u3002\u3053\u306e\u8a9e\u5f59\u898f\u5247\u3092\u4f7f\u7528\u3059\u308b\u3068\u3001\u524d\u306e\u7d50\u679c\u306f\u6b21\u306e\u3088\u3046\u306b\u8868\u73fe\u3055\u308c\u307e\u3059\u3002\u6709\u754c\u53ef\u6e2c\u96c6\u5408\u306f\u3001\u5185\u90e8\u6e2c\u5b9a\u5024\u3068\u5916\u90e8\u6e2c\u5b9a\u5024\u304c\u4e00\u81f4\u3059\u308b\u6709\u754c\u96c6\u5408\u3067\u3059\u3002<\/p>

\u7121\u5236\u9650\u306e\u30bb\u30c3\u30c8\u306e\u5834\u5408\u306f\u3001\u7a7a\u9593\u3092\u57cb\u3081\u308b\u4e00\u9023\u306e\u30d6\u30ed\u30c3\u30af\u3092\u542b\u3080\u3001\u524d\u8ff0\u306e\u30b9\u30c6\u30fc\u30c8\u30e1\u30f3\u30c8\u3068\u540c\u69d8\u306e\u30b9\u30c6\u30fc\u30c8\u30e1\u30f3\u30c8\u3092\u4f5c\u6210\u3067\u304d\u307e\u3059\u3002<\/p>

\u524d\u306e\u30b9\u30c6\u30fc\u30c8\u30e1\u30f3\u30c8\u306e\u4e00\u822c\u5316<\/a><\/span><\/strong>\u2014<\/span> Let

$$ {A\\subset \\R^n} $$<\/div> \u3001 \u305d\u3057\u3066
$$ {(E_i)_{i\\geq0}} $$<\/div>\u51fa\u4f1a\u3044\u304c\u3042\u308b\u4e00\u9023\u306e\u6577\u77f3
$$ {\\R^n} $$<\/div> \u3002\u96c6\u5408A \u306f<\/i><\/span>\u3001\u6b21\u306e\u5834\u5408\u306b\u306e\u307f\u30eb\u30d9\u30fc\u30b0\u53ef\u6e2c\u3067\u3059\u3002<\/p>
\u3059\u3079\u3066\u306e\u305f\u3081\u306b
$$ {i\\geq0,\\quad\\lambda_n^*(E_i\\cap A)+\\lambda_n^*(E_i\\setminus A)=\\mathrm{Vol}\\,(E_i)} $$<\/div> \u3002<\/center><\/div>

\u6e2c\u5b9a\u4e0d\u53ef\u80fd\u306a\u96c6\u5408<\/h2>

\u30ab\u30fc\u30c7\u30a3\u30ca\u30ea\u30c6\u30a3\u304b\u3089\u306f\u3001\u30eb\u30d9\u30fc\u30b0\u65cf\u304b\u3069\u3046\u304b\u3092\u5224\u65ad\u3059\u308b\u3053\u3068\u306f\u3067\u304d\u307e\u305b\u3093\u3002

$$ {\\R^n} $$<\/div>\u306e\u3059\u3079\u3066\u306e\u90e8\u5206\u306e\u96c6\u5408\u306b\u7b49\u3057\u3044\u304b\u7b49\u3057\u304f\u306a\u3044
$$ {\\R^n} $$<\/div> : \u3053\u308c\u3089 2 \u3064\u306e\u90e8\u5206\u30bb\u30c3\u30c8\u306e\u305d\u308c\u305e\u308c\u306f\u540c\u3058\u57fa\u6570\u3092\u6301\u3061\u307e\u3059\u3002
$$ {2^{\\mathfrak c}} $$<\/div> \u3002<\/p>

\u79c1\u305f\u3061\u306f\u6e2c\u5b9a\u4e0d\u53ef\u80fd\u306a\u96c6\u5408\u306e\u4f8b\u3092\u77e5\u3063\u3066\u3044\u307e\u3059\u3002\u6700\u3082\u5358\u7d14\u306a\u3082\u306e\u306e 1 \u3064\u306f\u30011905 \u5e74\u306b Giuseppe Vitali \u306b\u3088\u3063\u3066\u767a\u660e\u3055\u308c\u305f Vitali \u96c6\u5408\u3067\u3059\u3002

$$ {\\R\/\\Q} $$<\/div>\u3059\u3079\u3066[0,1] \u306e<\/span>\u7bc4\u56f2\u3067\u9078\u629e\u3055\u308c\u307e\u3059\u3002\u3082\u3046 1 \u3064\u306e\u7d20\u6674\u3089\u3057\u3044\u4f8b\u306f\u3001\u6b21\u306e\u5358\u4f4d\u30dc\u30fc\u30eb<\/a><\/span>\u306e\u30b5\u30d6\u30bb\u30c3\u30c8<\/a><\/span>\u3067\u3059\u3002
$$ {\\R^3} $$<\/div>\u3053\u308c\u306f\u30d0\u30ca\u30c3\u30cf\u30fb\u30bf\u30eb\u30b9\u30ad\u306e\u30d1\u30e9\u30c9\u30c3\u30af\u30b9\u3092\u5f15\u304d\u8d77\u3053\u3057\u307e\u3059\u3002<\/p>

\u3053\u308c\u3089\u306e\u4f8b\u306f\u3069\u3061\u3089\u3082\u9078\u629e\u306e\u516c\u7406<\/a><\/span>\u306b\u8a34\u3048\u307e\u3059\u3002\u3053\u308c\u306f\u5076\u7136\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002 Robert M. Solovay \u304c 1970 \u5e74\u306b\u767a\u8868\u3057\u305fSolovay \u30e2\u30c7\u30eb<\/span>\u306e\u5b58\u5728\u306f\u3001<\/span>\u5b9f\u969b\u3001\u9078\u629e\u516c\u7406\u306e\u306a\u3044 ZF\u96c6\u5408\u8ad6<\/a><\/span>\u3067\u306f\u3001\u6e2c\u5b9a\u4e0d\u53ef\u80fd\u306a\u96c6\u5408\u306e\u5b58\u5728\u3092\u8a3c\u660e\u3059\u308b\u3053\u3068\u306f\u671f\u5f85\u3067\u304d\u306a\u3044\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u307e\u3059 (\u3055\u3089\u306b\u3001\u3053\u308c\u306f\u516c\u7406\u3092\u4eee\u5b9a\u3057\u3066\u3082)\u4f9d\u5b58\u7684\u306a\u9078\u629e\u3067\u3059\uff09\u3002<\/p><\/div>

\n\"\u30eb\u30d9\u30fc\u30b0\u90e8\u65cf<\/figure>

\u53c2\u8003\u8cc7\u6599<\/h2>
  1. Lebesgue sigma-algebra \u2013 anglais<\/a><\/li>
  2. Sigma-\u00e1lgebra de Lebesgue \u2013 portugais<\/a><\/li>
  3. Tribu \u2013 catalan<\/a><\/li>
  4. Tribu \u2013 allemand<\/a><\/li>
  5. Tribu \u2013 anglais<\/a><\/li>
  6. Tribu (desambiguaci\u00f3n) \u2013 espagnol<\/a><\/li><\/ol><\/div>\n
    \n

    \n \u30eb\u30d9\u30fc\u30b0\u90e8\u65cf – \u5b9a\u7fa9\u30fb\u95a2\u9023\u52d5\u753b\n <\/h2>\n
    \n \n
    \n
    \n