{"id":34042,"date":"2021-03-12T00:00:39","date_gmt":"2021-03-12T00:00:39","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=34042"},"modified":"2025-08-15T08:28:55","modified_gmt":"2025-08-15T08:28:55","slug":"quid-est-sigma-algebra-definitiones-et-exempla","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/la\/quid-est-sigma-algebra-definitiones-et-exempla\/","title":{"rendered":"Quid est sigma-algebra? Definitiones et Exempla"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Quid est sigma-algebra? Definitio et exempla<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>Summarium<\/strong><br \/><em>In hac lectione tractatur momentum sigma-algebrae in theoria probabilitatis. Sigma-algebra est structura quae omnes eventus mensurabiles spatii exemplaris continet, permittens definire mensuram probabilitatis. Per exempla practica, ut iactus nummorum et tempus vitae machinae electronicae, explicatur quomodo sigma-algebra ex partibus spatii exemplaris construatur. Item exhibetur sigma-algebra Borelliana, spatio exemplari continuo associata, et explicantur eius eventus borelliani.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>PROPOSITA DISCENDI:<\/strong><br \/>\nHis peractis, discipulus poterit:\n<\/p>\n<ol>\n<li><strong>Intelligere<\/strong> definitionem et proprietates Sigma-Algebrae, tamquam structurae mathematicae quae permittit definire mensuram probabilitatis.<\/li>\n<li><strong>Identificare<\/strong> elementa quae Sigma-Algebra componunt eiusque relationem cum eventibus mensurabilibus spatii exemplaris.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>INDEX CONTENTORUM<\/strong><br \/>\n<a href=\"#1\">DEFINITIO SIGMA-ALGEBRAE<\/a><br \/>\n<a href=\"#2\">SIGMA-ALGEBRA IN IACTIBUS NUMMORUM<\/a><br \/>\n<a href=\"#3\">SIGMA-ALGEBRAE IN CASIBUS CONTINUIS<\/a><\/p>\n<p><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;\"><a href=\"https:\/\/www.youtube.com\/watch?v=cBnDYSiuyfE&amp;t=121s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">Eventus mensurabiles in spatio probabilitatum per sigma-algebram apparent.<\/span><\/strong><\/a> Per hanc notionem, conceptus initio intuitivus in structuram formaliter mathematicam mutatur, quae permittet definire mensuram probabilitatis.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Definitio Sigma-Algebrae<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=cBnDYSiuyfE&amp;t=209s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">Sigma-Algebra <span class=\"katex-eq\" data-katex-display=\"false\">\\Sigma<\/span> (sive \u03c3-algebra) est structura quae omnes eventus mensurabiles spatii exemplaris continet.<\/span><\/strong><\/a> Dicitur par <span class=\"katex-eq\" data-katex-display=\"false\">\\Sigma_{\\Omega} = (\\Omega, \\mathcal{A}_{\\Omega})<\/span> \u03c3-algebra <strong><a href=\"https:\/\/toposuranos.com\/el-espacio-de-probabilidades-el-espacio-muestral\/\" rel=\"noopener\" target=\"_blank\">spatii exemplaris<\/a><\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span> esse, si impletur:\n<\/p>\n<ol style=\"text-align: justify;\">\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;\">Omnia obiecta <span class=\"katex-eq\" data-katex-display=\"false\">E\\in\\mathcal{A}_\\Omega<\/span> vocantur <strong>Eventus <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span>.<\/strong><\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Sigma-algebra in iactibus nummorum<\/h2>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: justify; color: #008000; background-color: #dddddd;\"><strong>EXEMPLUM 1<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=cBnDYSiuyfE&amp;t=321s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">Pro iactu unius nummi,<\/span><\/strong><\/a> \u03c3-algebra datur per <span class=\"katex-eq\" data-katex-display=\"false\">\\Sigma_{1m}=(\\Omega_{1m}, \\mathcal{A}_{1m})<\/span>, ubi<\/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<p>Quisque elementum <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}_{1m}<\/span> est eventus qui sic identificatur:<\/p>\n<ul>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\emptyset<\/span> \u00abNec caput nec signum cadit\u00bb (eventus impossibilis est).<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{C\\}<\/span> \u00abEventus est quo caput exit\u00bb.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{S\\}<\/span> \u00abEventus est quo signum exit\u00bb.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{1m}= \\{C,S\\}<\/span> \u00abCadit alterutrum, sive caput sive signum\u00bb (eventus certus est).<\/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>EXEMPLUM 2<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=cBnDYSiuyfE&amp;t=558s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">Si pro una, duos nummos iaciamus,<\/span><\/strong><\/a> \u03c3-algebra possibilis <span class=\"katex-eq\" data-katex-display=\"false\">\\Sigma_{2m}=(\\Omega_{2m}, \\mathcal{A}_{2m})<\/span> obtineri potest ex partibus <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{2m}<\/span>. Hoc modo habemus:<\/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<p>Quisque elementum <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}_{2m}<\/span> est eventus <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{2m}<\/span>. Infra nonnulli eorum nominantur:<\/p>\n<ul>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\emptyset<\/span> \u00abNullus exitus habetur\u00bb (eventus impossibilis est).<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{(C,C)\\}<\/span> \u00abBis caput continuo cadit\u00bb.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{(C,S)\\}<\/span> \u00abPrimum caput, deinde signum exit\u00bb.<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> \u00abPrimum est caput, secundum quidlibet\u00bb.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{(C,C);(S,C)\\}<\/span> \u00abPrimum quidlibet est, secundum autem caput\u00bb.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{(C,C);(S,S)\\}<\/span> \u00abUterque iactus eundem exitum dat\u00bb.<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> \u00abSi primum est signum, secundum quoque signum est; aliter secundum quidlibet est\u00bb.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{(C,C);(S,C);(S,S)\\}<\/span> \u00abSi primum est caput, secundum quoque caput est; aliter secundum quidlibet est\u00bb.<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> \u00abQuilibet exitus possibilis obtinetur\u00bb (eventus certus).<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Sigma-algebrae in casibus continuis<\/h2>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: justify; color: #008000; background-color: #dddddd;\"><strong>EXEMPLUM 3<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=cBnDYSiuyfE&amp;t=1265s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">Pro tempore vitae (horis mensurato) instrumenti electronici quod quovis momento laedi potest,<\/span><\/strong><\/a> \u03c3-algebra <span class=\"katex-eq\" data-katex-display=\"false\">\\Sigma_e = (\\Omega_e, \\mathcal{A}_e)<\/span> dabitur per<\/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>Ita intervalla <span class=\"katex-eq\" data-katex-display=\"false\">I_t = ]0,t[\\in\\mathcal{A}_e <\/span> interpretari possunt ut \u00abinstrumentum electronicum recte operatur per intervallum <span class=\"katex-eq\" data-katex-display=\"false\">t<\/span> horarum continuatarum usque dum corrumpitur\u00bb.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Sigma-algebra probabilitatum spatio exemplari continuo associata etiam <strong>\u03c3-algebra Borelliana<\/strong> appellatur, eiusque eventus borelliani vocantur.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Quid est sigma-algebra? Definitio et exempla SummariumIn hac lectione tractatur momentum sigma-algebrae in theoria probabilitatis. Sigma-algebra est structura quae omnes eventus mensurabiles spatii exemplaris continet, permittens definire mensuram probabilitatis. Per exempla practica, ut iactus nummorum et tempus vitae machinae electronicae, explicatur quomodo sigma-algebra ex partibus spatii exemplaris construatur. Item exhibetur sigma-algebra Borelliana, spatio exemplari continuo [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":25427,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":76,"footnotes":""},"categories":[1298,1364],"tags":[],"class_list":["post-34042","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematica","category-probabilitates-et-statistica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Quid est sigma-algebra? Definitiones et Exempla - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Perspicere essentiam sigma-algebrarum in theoria probabilitatis per hanc accuratam ducem. 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