{"id":25433,"date":"2021-03-12T00:00:01","date_gmt":"2021-03-12T00:00:01","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=25433"},"modified":"2024-06-05T04:20:00","modified_gmt":"2024-06-05T04:20:00","slug":"o-que-e-uma-sigma-algebra-definicoes-e-exemplos","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/pt\/o-que-e-uma-sigma-algebra-definicoes-e-exemplos\/","title":{"rendered":"O que \u00e9 uma sigma-\u00e1lgebra? Defini\u00e7\u00f5es e Exemplos"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>O que \u00e9 uma sigma-\u00e1lgebra? Defini\u00e7\u00e3o e exemplos<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>Resumo<\/strong><br \/><em>Nesta aula aborda-se a import\u00e2ncia da sigma-\u00e1lgebra na teoria da probabilidade. A sigma-\u00e1lgebra \u00e9 uma estrutura que cont\u00e9m todos os eventos mensur\u00e1veis de um espa\u00e7o amostral, permitindo definir uma medida de probabilidade. Atrav\u00e9s de exemplos pr\u00e1ticos, como os lan\u00e7amentos de moedas e o tempo de vida de um aparelho eletr\u00f4nico, explica-se como a sigma-\u00e1lgebra \u00e9 constru\u00edda a partir das partes do espa\u00e7o amostral. Tamb\u00e9m \u00e9 apresentada a sigma-\u00e1lgebra de Borel, associada a um espa\u00e7o amostral cont\u00ednuo, e s\u00e3o explicados seus eventos borelianos.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>OBJETIVOS DE APRENDIZADO:<\/strong><br \/>\nAo completar esta aula, o aluno ser\u00e1 capaz de:\n<\/p>\n<ol>\n<li><strong>Compreender<\/strong> a defini\u00e7\u00e3o e caracter\u00edsticas de uma sigma-\u00e1lgebra, como estrutura matem\u00e1tica que permite definir uma medida de probabilidade.<\/li>\n<li><strong>Identificar<\/strong> os elementos que comp\u00f5em uma sigma-\u00e1lgebra e sua rela\u00e7\u00e3o com os eventos mensur\u00e1veis de um espa\u00e7o amostral.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>\u00cdNDICE DE CONTE\u00daDOS<\/strong><br \/>\n<a href=\"#1\">DEFINI\u00c7\u00c3O DE UMA SIGMA-\u00c1LGEBRA<\/a><br \/>\n<a href=\"#2\">A SIGMA-\u00c1LGEBRA NOS LAN\u00c7AMENTOS DE MOEDAS<\/a><br \/>\n<a href=\"#3\">A SIGMA-\u00c1LGEBRA EM CASOS CONT\u00cdNUOS<\/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;\">Os eventos mensur\u00e1veis surgem no espa\u00e7o de probabilidades atrav\u00e9s da sigma-\u00e1lgebra.<\/span><\/strong><\/a> Atrav\u00e9s desta ideia, transforma-se uma no\u00e7\u00e3o inicialmente intuitiva em uma estrutura formalmente matem\u00e1tica que permitir\u00e1 definir uma medida de probabilidade.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Defini\u00e7\u00e3o de uma Sigma-\u00c1lgebra<\/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;\">Uma Sigma-\u00c1lgebra <span class=\"katex-eq\" data-katex-display=\"false\">\\Sigma<\/span> (ou \u03c3-\u00e1lgebra) \u00e9 uma estrutura que cont\u00e9m todos os eventos mensur\u00e1veis de um espa\u00e7o amostral.<\/span><\/strong><\/a> Diz-se que o par <span class=\"katex-eq\" data-katex-display=\"false\">\\Sigma_{\\Omega} = (\\Omega, \\mathcal{A}_{\\Omega})<\/span> \u00e9 uma \u03c3-\u00e1lgebra de um <strong><a href=\"https:\/\/toposuranos.com\/el-espacio-de-probabilidades-el-espacio-muestral\/\" rel=\"noopener\" target=\"_blank\">espa\u00e7o amostral<\/a><\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span> se satisfaz as seguintes condi\u00e7\u00f5es:<\/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;\">Todos os objetos <span class=\"katex-eq\" data-katex-display=\"false\">E\\in\\mathcal{A}_\\Omega<\/span> s\u00e3o chamados de <strong>Eventos de <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span>.<\/strong><\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>A sigma-\u00e1lgebra nos lan\u00e7amentos de moedas<\/h2>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: justify; color: #008000; background-color: #dddddd;\"><strong>EXEMPLO 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;\">Para o lan\u00e7amento de uma moeda,<\/span><\/strong><\/a> a sigma-\u00e1lgebra \u00e9 dada por <span class=\"katex-eq\" data-katex-display=\"false\">\\Sigma_{1m}=(\\Omega_{1m}, \\mathcal{A}_{1m})<\/span>, onde<\/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>Cada elemento de <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}_{1m}<\/span> \u00e9 um evento que pode ser identificado como:<\/p>\n<ul>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\emptyset<\/span> \u00abN\u00e3o sai cara nem coroa\u00bb (\u00e9 o evento imposs\u00edvel).<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{C\\}<\/span> \u00ab\u00c9 o evento em que sai cara\u00bb.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{S\\}<\/span> \u00ab\u00c9 o evento em que sai coroa\u00bb.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{1m}= \\{C,S\\}<\/span> \u00abSai qualquer uma das duas, cara ou coroa\u00bb (\u00e9 o evento certo).<\/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>EXEMPLO 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;\">Se, em vez de uma, lan\u00e7amos duas moedas,<\/span><\/strong><\/a> uma poss\u00edvel \u03c3-\u00e1lgebra <span class=\"katex-eq\" data-katex-display=\"false\">\\Sigma_{2m}=(\\Omega_{2m}, \\mathcal{A}_{2m})<\/span> pode ser obtida a partir das partes de <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{2m}<\/span>. Dessa forma, temos o seguinte:<\/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>Cada elemento de <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}_{2m}<\/span> \u00e9 um evento de <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{2m}<\/span>. A seguir, s\u00e3o nomeados alguns deles:<\/p>\n<ul>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\emptyset<\/span> \u00abN\u00e3o h\u00e1 nenhum resultado\u00bb (evento imposs\u00edvel).<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">{(C,C)}<\/span> \u00abDuas caras consecutivas\u00bb.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">{(C,S)}<\/span> \u00abPrimeiro sai cara e depois coroa\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> \u00abO primeiro \u00e9 cara e o segundo pode ser qualquer coisa\u00bb.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">{(C,C);(S,C)}<\/span> \u00abO primeiro pode ser qualquer coisa e o segundo \u00e9 cara\u00bb.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">{(C,C);(S,S)}<\/span> \u00abAmbos os lan\u00e7amentos d\u00e3o o mesmo resultado\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> \u00abSe o primeiro for coroa, o segundo tamb\u00e9m \u00e9 coroa, caso contr\u00e1rio, o segundo pode ser qualquer coisa\u00bb.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">{(C,C);(S,C);(S,S)}<\/span> \u00abSe o primeiro for cara, o segundo tamb\u00e9m \u00e9 cara, caso contr\u00e1rio, o segundo pode ser qualquer coisa\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> \u00abQualquer resultado \u00e9 poss\u00edvel\u00bb (evento certo).<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"3\"><\/a><\/p>\n<h2>As sigma-\u00e1lgebras em casos cont\u00ednuos<\/h2>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: justify; color: #008000; background-color: #dddddd;\"><strong>EXEMPLO 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;\">Para o tempo de vida (medido em horas) de um aparelho eletr\u00f4nico que pode estragar a qualquer momento,<\/span><\/strong><\/a> a \u03c3-\u00e1lgebra <span class=\"katex-eq\" data-katex-display=\"false\">\\Sigma_e = (\\Omega_e, \\mathcal{A}_e)<\/span> ser\u00e1 dada por:<\/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>Dessa forma, os intervalos <span class=\"katex-eq\" data-katex-display=\"false\">I_t = ]0,t[\\in\\mathcal{A}_e <\/span> podem ser interpretados como \u00abo aparelho eletr\u00f4nico funciona corretamente durante um intervalo de <span class=\"katex-eq\" data-katex-display=\"false\">t<\/span> horas seguidas at\u00e9 que estrague.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">A sigma-\u00e1lgebra de probabilidade associada a um espa\u00e7o amostral cont\u00ednuo tamb\u00e9m \u00e9 conhecida como <strong>\u03c3-\u00e1lgebra de Borel<\/strong> e seus eventos s\u00e3o conhecidos como borelianos.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>O que \u00e9 uma sigma-\u00e1lgebra? Defini\u00e7\u00e3o e exemplos ResumoNesta aula aborda-se a import\u00e2ncia da sigma-\u00e1lgebra na teoria da probabilidade. A sigma-\u00e1lgebra \u00e9 uma estrutura que cont\u00e9m todos os eventos mensur\u00e1veis de um espa\u00e7o amostral, permitindo definir uma medida de probabilidade. Atrav\u00e9s de exemplos pr\u00e1ticos, como os lan\u00e7amentos de moedas e o tempo de vida de [&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":103,"footnotes":""},"categories":[571,672],"tags":[],"class_list":["post-25433","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-matematica-pt","category-probabilidades-e-estatistica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>O que \u00e9 uma sigma-\u00e1lgebra? 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