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?

  • 2
    $\begingroup$ 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 '16 at 20:56

$\Sigma_1 = \{\emptyset, \{a\}, \{b, c\}, \{a, b, c\}\}$ 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.