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The class $\mathscr{S}\subset \mathcal{P}(\Omega)$ is a semi-algebra,and $\sigma(\mathscr{S})$ is the $\sigma$-algebra generated by $\mathscr{S}$. Here is a proposition:if $\Omega$ is countable,then $$\sigma(\mathscr{S})=\{\sum_{n=1}^{\infty}A_n:\forall n\geqslant1,A_n\in\mathscr{S}\}.$$ I want to know whether the proposition is right?If the proposition is right could you please provide me a proof of it.Thank you!

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    $\begingroup$ What does the sum symbol stand for? $\endgroup$
    – Julien
    Commented Oct 7, 2017 at 15:00
  • $\begingroup$ The sum symbol stands for the merge of mutually disjoint sets. $\endgroup$
    – J.Yu
    Commented Oct 7, 2017 at 15:11
  • $\begingroup$ What is the merge of sets? Do you refer to the union? $\endgroup$ Commented Oct 7, 2017 at 15:45
  • $\begingroup$ Yeah,sorry ,I mean the union. $\endgroup$
    – J.Yu
    Commented Oct 8, 2017 at 0:19

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I believe your proposition does not hold.

Consider the semi-algebra over $\Omega = \mathbb{N}$ given by:

$\mathbb{N}$, $\emptyset$, {any single even number in $\mathbb{N}$}, $\mathbb{N}-$any finite number of even numbers in $\mathbb{N}$

The set of all odd numbers in $\mathbb{N}$ should belong to the generated $\sigma$-algebra (as the countable intersection of the sets $E_{n}=$[$\mathbb{N}-$all even numbers in $[-n;n]$]) but cannot be written as a countable union of sets belonging to the semi-algebra.

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  • $\begingroup$ Thank you for your reply.And I get it now. $\endgroup$
    – J.Yu
    Commented Oct 8, 2017 at 11:10

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