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Math Help - Sigma Algebra Exercise

  1. #1
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    Sigma Algebra Exercise

    Hello

    Could you plese help me on this exercise?

    Let F1 and F2 two sigma algebras of subsets of  \Omega and F1\subseteq{F2}
    Prove that F1\cup{F2} is a sigma algebra.
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  2. #2
    Super Member girdav's Avatar
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    Re: Sigma Algebra Exercise

    Apply the definition. Show that :
    (i) \Omega\in F_1\cup F_2;
    (ii) If A\in F_1\cup F_2, show that \complement_{\Omega}A\in F_1\cup F_2;
    (iii) Let \{A_n\}_{n\in\mathbb N}\in F_1\cup F_2. Show that \bigcup_{n\in\mathbb N}A_n\in F_1\cup F_2.
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    Re: Sigma Algebra Exercise

    Another way: F_1\cup F_2=F_2 which by hypothesis is \sigma-algebra.
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  4. #4
    Super Member girdav's Avatar
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    Re: Sigma Algebra Exercise

    We notice that the hypothesis F_1\subset F_2 (or F_2\subset F_1) is necessary: if we consider \Omega=\left\{1;2;3\right\}, F_1 =\left\{\emptyset, \left\{1\right\},\left\{2;3\right\},\Omega\right\} and F_2 =\left\{\emptyset, \left\{2\right\},\left\{1;3\right\},\Omega\right\} then F_1\cup F_2 =\left\{\emptyset, \left\{1\right\},\left\{2\right\}; \left\{1;3\right\};\left\{2;3\right\},\Omega\right  \} which is not a \sigma-algebra since \left\{1\right\}\cup \left\{2\right\}\notin F_1\cup F_2.
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