{"id":25447,"date":"2021-03-12T00:00:50","date_gmt":"2021-03-12T00:00:50","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=25447"},"modified":"2024-06-05T04:21:42","modified_gmt":"2024-06-05T04:21:42","slug":"quest-ce-quune-sigma-algebre-definitions-et-exemples","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/","title":{"rendered":"Qu&#8217;est-ce qu&#8217;une sigma-alg\u00e8bre ? D\u00e9finitions et Exemples"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Qu&#8217;est-ce qu&#8217;une sigma-alg\u00e8bre ? D\u00e9finition et exemples<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>R\u00e9sum\u00e9<\/strong><br \/><em>Dans ce cours, nous abordons l&#8217;importance de la sigma-alg\u00e8bre dans la th\u00e9orie des probabilit\u00e9s. La sigma-alg\u00e8bre est une structure contenant tous les \u00e9v\u00e9nements mesurables d&#8217;un espace d&#8217;\u00e9chantillonnage, permettant de d\u00e9finir une mesure de probabilit\u00e9. \u00c0 travers des exemples pratiques, tels que les lancers de pi\u00e8ces de monnaie et la dur\u00e9e de vie d&#8217;un appareil \u00e9lectronique, nous expliquons comment la sigma-alg\u00e8bre est construite \u00e0 partir des parties de l&#8217;espace d&#8217;\u00e9chantillonnage. Nous pr\u00e9sentons \u00e9galement la sigma-alg\u00e8bre de Borel, associ\u00e9e \u00e0 un espace d&#8217;\u00e9chantillonnage continu, et expliquons ses \u00e9v\u00e9nements bor\u00e9liens.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>OBJECTIFS D&#8217;APPRENTISSAGE :<\/strong><br \/>\n\u00c0 la fin de ce cours, l&#8217;\u00e9tudiant sera capable de :\n<\/p>\n<ol>\n<li><strong>Comprendre<\/strong> la d\u00e9finition et les caract\u00e9ristiques d&#8217;une Sigma-Alg\u00e8bre, comme structure math\u00e9matique permettant de d\u00e9finir une mesure de probabilit\u00e9.<\/li>\n<li><strong>Identifier<\/strong> les \u00e9l\u00e9ments constituant une Sigma-Alg\u00e8bre et leur relation avec les \u00e9v\u00e9nements mesurables d&#8217;un espace d&#8217;\u00e9chantillonnage.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>SOMMAIRE<\/strong><br \/>\n<a href=\"#1\">D\u00c9FINITION D&#8217;UNE SIGMA-ALG\u00c8BRE<\/a><br \/>\n<a href=\"#2\">LA SIGMA-ALG\u00c8BRE DANS LES LANCERS DE PI\u00c8CES<\/a><br \/>\n<a href=\"#3\">LES SIGMA-ALG\u00c8BRES DANS LES CAS CONTINUS<\/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; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=cBnDYSiuyfE&amp;t=121s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">Les \u00e9v\u00e9nements mesurables apparaissent dans l&#8217;espace des probabilit\u00e9s \u00e0 travers la sigma-alg\u00e8bre.<\/span><\/strong><\/a> Gr\u00e2ce \u00e0 cette id\u00e9e, une notion initialement intuitive est transform\u00e9e en une structure formellement math\u00e9matique qui permet de d\u00e9finir une mesure de probabilit\u00e9.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>D\u00e9finition d&#8217;une Sigma-Alg\u00e8bre<\/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;\">Une Sigma-Alg\u00e8bre <span class=\"katex-eq\" data-katex-display=\"false\">\\Sigma<\/span> (ou \u03c3-alg\u00e8bre) est une structure qui contient tous les \u00e9v\u00e9nements mesurables d&#8217;un espace d&#8217;\u00e9chantillonnage.<\/span><\/strong><\/a> On dit que la paire <span class=\"katex-eq\" data-katex-display=\"false\">\\Sigma_{\\Omega} = (\\Omega, \\mathcal{A}_{\\Omega})<\/span> est une \u03c3-alg\u00e8bre d&#8217;un <strong><a href=\"https:\/\/toposuranos.com\/el-espacio-de-probabilidades-el-espacio-muestral\/\" rel=\"noopener\" target=\"_blank\">espace d&#8217;\u00e9chantillonnage<\/a><\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span> si elle satisfait aux conditions suivantes :<\/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;\">Tous les objets <span class=\"katex-eq\" data-katex-display=\"false\">E\\in\\mathcal{A}_\\Omega<\/span> sont appel\u00e9s <strong>\u00c9v\u00e9nements de <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span>.<\/strong><\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>La sigma-alg\u00e8bre dans les lancers de pi\u00e8ces<\/h2>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: justify; color: #008000; background-color: #dddddd;\"><strong>EXEMPLE 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;\">Pour le lancer d&#8217;une pi\u00e8ce,<\/span><\/strong><\/a> la \u03c3-alg\u00e8bre est donn\u00e9e par <span class=\"katex-eq\" data-katex-display=\"false\">\\Sigma_{1m}=(\\Omega_{1m}, \\mathcal{A}_{1m})<\/span>, o\u00f9<\/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>Chaque \u00e9l\u00e9ment de <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}_{1m}<\/span> est un \u00e9v\u00e9nement qui est identifi\u00e9 de la mani\u00e8re suivante :<\/p>\n<ul>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\emptyset<\/span> \u00abNi face ni pile\u00bb (c&#8217;est l&#8217;\u00e9v\u00e9nement impossible).<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{C\\}<\/span> \u00abC&#8217;est l&#8217;\u00e9v\u00e9nement o\u00f9 la face appara\u00eet\u00bb.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{S\\}<\/span> \u00abC&#8217;est l&#8217;\u00e9v\u00e9nement o\u00f9 le pile appara\u00eet\u00bb.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{1m}= \\{C,S\\}<\/span> \u00abSoit la face soit le pile appara\u00eet\u00bb (c&#8217;est l&#8217;\u00e9v\u00e9nement certain).<\/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>EXEMPLE 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;\">Si au lieu d&#8217;une, nous lan\u00e7ons deux pi\u00e8ces,<\/span><\/strong><\/a> une possible \u03c3-alg\u00e8bre <span class=\"katex-eq\" data-katex-display=\"false\">\\Sigma_{2m}=(\\Omega_{2m}, \\mathcal{A}_{2m})<\/span> peut \u00eatre obtenue \u00e0 partir des parties de <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{2m}<\/span>. Ainsi nous avons :<\/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>Chaque \u00e9l\u00e9ment de <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}_{2m}<\/span> est un \u00e9v\u00e9nement de <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{2m}<\/span>. Voici quelques-uns d&#8217;entre eux :<\/p>\n<ul>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\emptyset<\/span> \u00abAucun r\u00e9sultat obtenu\u00bb (c&#8217;est l&#8217;\u00e9v\u00e9nement impossible).<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{(C,C)\\}<\/span> \u00abLa face appara\u00eet deux fois de suite\u00bb.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{(C,S)\\}<\/span> \u00abLa face appara\u00eet d&#8217;abord, puis le pile\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> \u00abLe premier est la face, le second peut \u00eatre n&#8217;importe quoi\u00bb.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{(C,C);(S,C)\\}<\/span> \u00abLe premier peut \u00eatre n&#8217;importe quoi, le second est la face\u00bb.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{(C,C);(S,S)\\}<\/span> \u00abLes deux lancers donnent le m\u00eame r\u00e9sultat\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 le premier est le pile, le second est aussi le pile, sinon le second peut \u00eatre n&#8217;importe quoi\u00bb.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{(C,C);(S,C);(S,S)\\}<\/span> \u00abSi le premier est la face, le second est aussi la face, sinon le second peut \u00eatre n&#8217;importe quoi\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> \u00abN&#8217;importe quel r\u00e9sultat est obtenu\u00bb (\u00e9v\u00e9nement certain).<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Les sigma-alg\u00e8bres dans les cas continus<\/h2>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: justify; color: #008000; background-color: #dddddd;\"><strong>EXEMPLE 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;\">Pour la dur\u00e9e de vie (mesur\u00e9e en heures) d&#8217;un appareil \u00e9lectronique qui pourrait se d\u00e9t\u00e9riorer \u00e0 tout moment,<\/span><\/strong><\/a> la \u03c3-alg\u00e8bre <span class=\"katex-eq\" data-katex-display=\"false\">\\Sigma_e = (\\Omega_e, \\mathcal{A}_e)<\/span> est d\u00e9finie par<\/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>Ainsi, les intervalles <span class=\"katex-eq\" data-katex-display=\"false\">I_t = ]0,t[\\in\\mathcal{A}_e <\/span> peuvent \u00eatre interpr\u00e9t\u00e9s comme \u00abl&#8217;appareil \u00e9lectronique fonctionne correctement pendant une dur\u00e9e de <span class=\"katex-eq\" data-katex-display=\"false\">t<\/span> heures cons\u00e9cutives jusqu&#8217;\u00e0 ce qu&#8217;il tombe en panne.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">La sigma-alg\u00e8bre de probabilit\u00e9s associ\u00e9e \u00e0 un espace d&#8217;\u00e9chantillonnage continu est \u00e9galement connue sous le nom de <strong>\u03c3-alg\u00e8bre de Borel<\/strong>, et ses \u00e9v\u00e9nements sont appel\u00e9s bor\u00e9liens.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Qu&#8217;est-ce qu&#8217;une sigma-alg\u00e8bre ? D\u00e9finition et exemples R\u00e9sum\u00e9Dans ce cours, nous abordons l&#8217;importance de la sigma-alg\u00e8bre dans la th\u00e9orie des probabilit\u00e9s. La sigma-alg\u00e8bre est une structure contenant tous les \u00e9v\u00e9nements mesurables d&#8217;un espace d&#8217;\u00e9chantillonnage, permettant de d\u00e9finir une mesure de probabilit\u00e9. \u00c0 travers des exemples pratiques, tels que les lancers de pi\u00e8ces de monnaie et [&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":102,"footnotes":""},"categories":[569,682],"tags":[],"class_list":["post-25447","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematiques","category-probabilites-et-statistiques"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Qu&#039;est-ce qu&#039;une sigma-alg\u00e8bre ? D\u00e9finitions et Exemples - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"D\u00e9couvrez l&#039;essence des sigma-alg\u00e8bres dans la th\u00e9orie des probabilit\u00e9s avec ce guide d\u00e9taill\u00e9. Apprenez comment sont d\u00e9finis les \u00e9v\u00e9nements mesurables.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Qu&#039;est-ce qu&#039;une sigma-alg\u00e8bre ? D\u00e9finitions et Exemples\" \/>\n<meta property=\"og:description\" content=\"D\u00e9couvrez l&#039;essence des sigma-alg\u00e8bres dans la th\u00e9orie des probabilit\u00e9s avec ce guide d\u00e9taill\u00e9. Apprenez comment sont d\u00e9finis les \u00e9v\u00e9nements mesurables.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-03-12T00:00:50+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-06-05T04:21:42+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/sigmaalgebras-1024x585.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Qu&#039;est-ce qu&#039;une sigma-alg\u00e8bre ? D\u00e9finitions et Exemples\" \/>\n<meta name=\"twitter:description\" content=\"D\u00e9couvrez l&#039;essence des sigma-alg\u00e8bres dans la th\u00e9orie des probabilit\u00e9s avec ce guide d\u00e9taill\u00e9. Apprenez comment sont d\u00e9finis les \u00e9v\u00e9nements mesurables.\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/sigmaalgebras.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"4 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Qu&#8217;est-ce qu&#8217;une sigma-alg\u00e8bre ? D\u00e9finitions et Exemples\",\"datePublished\":\"2021-03-12T00:00:50+00:00\",\"dateModified\":\"2024-06-05T04:21:42+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/\"},\"wordCount\":974,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/toposuranos.com\/material\/#organization\"},\"image\":{\"@id\":\"https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/sigmaalgebras.jpg\",\"articleSection\":[\"Math\u00e9matiques\",\"Probabilit\u00e9s et Statistiques\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/\",\"url\":\"https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/\",\"name\":\"Qu'est-ce qu'une sigma-alg\u00e8bre ? D\u00e9finitions et Exemples - toposuranos.com\/material\",\"isPartOf\":{\"@id\":\"https:\/\/toposuranos.com\/material\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/#primaryimage\"},\"image\":{\"@id\":\"https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/sigmaalgebras.jpg\",\"datePublished\":\"2021-03-12T00:00:50+00:00\",\"dateModified\":\"2024-06-05T04:21:42+00:00\",\"description\":\"D\u00e9couvrez l'essence des sigma-alg\u00e8bres dans la th\u00e9orie des probabilit\u00e9s avec ce guide d\u00e9taill\u00e9. Apprenez comment sont d\u00e9finis les \u00e9v\u00e9nements mesurables.\",\"breadcrumb\":{\"@id\":\"https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/#breadcrumb\"},\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/#primaryimage\",\"url\":\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/sigmaalgebras.jpg\",\"contentUrl\":\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/sigmaalgebras.jpg\",\"width\":1792,\"height\":1024},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Portada\",\"item\":\"https:\/\/toposuranos.com\/material\/es\/cursos-de-matematica-y-fisica\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Qu&rsquo;est-ce qu&rsquo;une sigma-alg\u00e8bre ? D\u00e9finitions et Exemples\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/toposuranos.com\/material\/#website\",\"url\":\"https:\/\/toposuranos.com\/material\/\",\"name\":\"toposuranos.com\/material\",\"description\":\"\",\"publisher\":{\"@id\":\"https:\/\/toposuranos.com\/material\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/toposuranos.com\/material\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"es\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/toposuranos.com\/material\/#organization\",\"name\":\"toposuranos.com\/material\",\"url\":\"https:\/\/toposuranos.com\/material\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"https:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/\",\"url\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png\",\"contentUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png\",\"width\":2400,\"height\":2059,\"caption\":\"toposuranos.com\/material\"},\"image\":{\"@id\":\"https:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/\"},\"sameAs\":[\"https:\/\/www.facebook.com\/groups\/toposuranos\",\"https:\/\/x.com\/topuranos\",\"https:\/\/www.youtube.com\/channel\/UC16yDm12cPcrwsE0fAM7X1g\",\"https:\/\/www.linkedin.com\/company\/69429190\"]},{\"@type\":\"Person\",\"@id\":\"https:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1\",\"name\":\"giorgio.reveco\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"https:\/\/toposuranos.com\/material\/#\/schema\/person\/image\/\",\"url\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg\",\"contentUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg\",\"caption\":\"giorgio.reveco\"},\"description\":\"Soy Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. Me dedico a desmitificar la f\u00edsica y las matem\u00e1ticas. Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.\",\"sameAs\":[\"http:\/\/toposuranos.com\/material\"],\"url\":\"https:\/\/toposuranos.com\/material\/author\/giorgio-reveco\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Qu'est-ce qu'une sigma-alg\u00e8bre ? D\u00e9finitions et Exemples - toposuranos.com\/material","description":"D\u00e9couvrez l'essence des sigma-alg\u00e8bres dans la th\u00e9orie des probabilit\u00e9s avec ce guide d\u00e9taill\u00e9. Apprenez comment sont d\u00e9finis les \u00e9v\u00e9nements mesurables.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/","og_locale":"es_ES","og_type":"article","og_title":"Qu'est-ce qu'une sigma-alg\u00e8bre ? D\u00e9finitions et Exemples","og_description":"D\u00e9couvrez l'essence des sigma-alg\u00e8bres dans la th\u00e9orie des probabilit\u00e9s avec ce guide d\u00e9taill\u00e9. Apprenez comment sont d\u00e9finis les \u00e9v\u00e9nements mesurables.","og_url":"https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/","og_site_name":"toposuranos.com\/material","article_publisher":"https:\/\/www.facebook.com\/groups\/toposuranos","article_published_time":"2021-03-12T00:00:50+00:00","article_modified_time":"2024-06-05T04:21:42+00:00","og_image":[{"url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/sigmaalgebras-1024x585.jpg","type":"","width":"","height":""}],"author":"giorgio.reveco","twitter_card":"summary_large_image","twitter_title":"Qu'est-ce qu'une sigma-alg\u00e8bre ? D\u00e9finitions et Exemples","twitter_description":"D\u00e9couvrez l'essence des sigma-alg\u00e8bres dans la th\u00e9orie des probabilit\u00e9s avec ce guide d\u00e9taill\u00e9. Apprenez comment sont d\u00e9finis les \u00e9v\u00e9nements mesurables.","twitter_image":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/sigmaalgebras.jpg","twitter_creator":"@topuranos","twitter_site":"@topuranos","twitter_misc":{"Escrito por":"giorgio.reveco","Tiempo de lectura":"4 minutos"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/#article","isPartOf":{"@id":"https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/"},"author":{"name":"giorgio.reveco","@id":"https:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1"},"headline":"Qu&#8217;est-ce qu&#8217;une sigma-alg\u00e8bre ? D\u00e9finitions et Exemples","datePublished":"2021-03-12T00:00:50+00:00","dateModified":"2024-06-05T04:21:42+00:00","mainEntityOfPage":{"@id":"https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/"},"wordCount":974,"commentCount":0,"publisher":{"@id":"https:\/\/toposuranos.com\/material\/#organization"},"image":{"@id":"https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/#primaryimage"},"thumbnailUrl":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/sigmaalgebras.jpg","articleSection":["Math\u00e9matiques","Probabilit\u00e9s et Statistiques"],"inLanguage":"es","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/","url":"https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/","name":"Qu'est-ce qu'une sigma-alg\u00e8bre ? D\u00e9finitions et Exemples - toposuranos.com\/material","isPartOf":{"@id":"https:\/\/toposuranos.com\/material\/#website"},"primaryImageOfPage":{"@id":"https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/#primaryimage"},"image":{"@id":"https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/#primaryimage"},"thumbnailUrl":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/sigmaalgebras.jpg","datePublished":"2021-03-12T00:00:50+00:00","dateModified":"2024-06-05T04:21:42+00:00","description":"D\u00e9couvrez l'essence des sigma-alg\u00e8bres dans la th\u00e9orie des probabilit\u00e9s avec ce guide d\u00e9taill\u00e9. Apprenez comment sont d\u00e9finis les \u00e9v\u00e9nements mesurables.","breadcrumb":{"@id":"https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/#breadcrumb"},"inLanguage":"es","potentialAction":[{"@type":"ReadAction","target":["https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/"]}]},{"@type":"ImageObject","inLanguage":"es","@id":"https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/#primaryimage","url":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/sigmaalgebras.jpg","contentUrl":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/sigmaalgebras.jpg","width":1792,"height":1024},{"@type":"BreadcrumbList","@id":"https:\/\/toposuranos.com\/material\/fr\/quest-ce-quune-sigma-algebre-definitions-et-exemples\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Portada","item":"https:\/\/toposuranos.com\/material\/es\/cursos-de-matematica-y-fisica\/"},{"@type":"ListItem","position":2,"name":"Qu&rsquo;est-ce qu&rsquo;une sigma-alg\u00e8bre ? D\u00e9finitions et Exemples"}]},{"@type":"WebSite","@id":"https:\/\/toposuranos.com\/material\/#website","url":"https:\/\/toposuranos.com\/material\/","name":"toposuranos.com\/material","description":"","publisher":{"@id":"https:\/\/toposuranos.com\/material\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/toposuranos.com\/material\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"es"},{"@type":"Organization","@id":"https:\/\/toposuranos.com\/material\/#organization","name":"toposuranos.com\/material","url":"https:\/\/toposuranos.com\/material\/","logo":{"@type":"ImageObject","inLanguage":"es","@id":"https:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png","width":2400,"height":2059,"caption":"toposuranos.com\/material"},"image":{"@id":"https:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.facebook.com\/groups\/toposuranos","https:\/\/x.com\/topuranos","https:\/\/www.youtube.com\/channel\/UC16yDm12cPcrwsE0fAM7X1g","https:\/\/www.linkedin.com\/company\/69429190"]},{"@type":"Person","@id":"https:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1","name":"giorgio.reveco","image":{"@type":"ImageObject","inLanguage":"es","@id":"https:\/\/toposuranos.com\/material\/#\/schema\/person\/image\/","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","caption":"giorgio.reveco"},"description":"Soy Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. Me dedico a desmitificar la f\u00edsica y las matem\u00e1ticas. Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.","sameAs":["http:\/\/toposuranos.com\/material"],"url":"https:\/\/toposuranos.com\/material\/author\/giorgio-reveco\/"}]}},"_links":{"self":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/25447","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/comments?post=25447"}],"version-history":[{"count":0,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/25447\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media\/25427"}],"wp:attachment":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media?parent=25447"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/categories?post=25447"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/tags?post=25447"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}