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Consider the following statement: If $\sum_{1}$, $\sum_{2}$ are $\sigma-$ algebras, then $\Sigma_{1}\cup\Sigma_{2}$ is in general not a $\sigma-$algbra.

In order to come up with an illustrative example of when this union is not a sigma algebra, I consider two very trivial sigma algebras:

Consider $\sum_{1}=\{\emptyset,S_{1}\}$ and $\Sigma_{2}=\{\emptyset,S_{2}\}$ then $\Sigma_{1}\cup\Sigma_{2}=\{\emptyset,S_{1},S_{2}\}$ .The union is by no means a sigma algebra- it does not contain the modified new sample space (presumably $S_{1}\cup S_{2}$) nor the fact that given that $S_{1}$ and $S_{2}$ are contained in the union, their union is not. Is this reasoning correct?

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    $\begingroup$ Usually when people talk about a "union of two sigma algebras" they mean two sigma algebras on the same set. $\endgroup$
    – Eric Wofsey
    Sep 15, 2016 at 20:56

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$\Sigma_1 = \{\emptyset, \{a\}, \{b, c\}, \{a, b, c\}\}$ and $\Sigma_2 = \{\emptyset, \{a, b\}, \{c\}, \{a, b, c\}\}$ is a trivial example of two $\sigma$-algebras on the same set whose union is not a $\sigma$-algebra: the union contains $\{a, b\}$ and $\{b, c\}$ but not their intersection.

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