Suppose $X$ is a nonempty set. If $\mathcal{E}\subseteq 2^{X}$ (the power set of $X$), the intersection of all $\sigma$-algebras containing $\mathcal{E}$ is called the $\sigma$-algebra generated by $\mathcal{E}$, and is denoted by $\mathcal{M}(\mathcal{E})$. I am trying to prove the following exercise:

If $\mathcal{M}(\mathcal{E})$ is the $\sigma$-algebra generated by $\mathcal{E}$, then $\mathcal{M}(\mathcal{E})$ is the union of the $\sigma$-algebras generated by $\mathcal{F}$ as $\mathcal{F}$ ranges over all countable subsets of $\mathcal{E}$. (Hint: Show that the latter object is a $\sigma$-algebra)

My attempt: Let $\mathcal{N}$ be the union of the $\sigma$-algebras generated by $\mathcal{F}$ as $\mathcal{F}$ ranges over all countable subsets of $\mathcal{E}$. I can prove that $\mathcal{N}$ is a $\sigma$-algebra.

But how do I prove that $\mathcal{E}\subseteq\mathcal{N}$?

If $\mathcal{E}$ is countable, then it is clear. But what happens if $\mathcal{E}$ is uncountable? I appreciate any help.

  • 1
    $\begingroup$ See here. $\endgroup$ – Michael Greinecker Dec 18 '13 at 22:23
  • 1
    $\begingroup$ This is probably $\mathcal E\subseteq2^X$, not $\mathcal E\subseteq X$. $\endgroup$ – Did Dec 18 '13 at 22:25
  • $\begingroup$ @Did: Corrected :-) $\endgroup$ – Prism Dec 18 '13 at 22:27
  • $\begingroup$ Actually, $\mathcal E\in2^X$ and $\mathcal E\subseteq X$ are equivalent hence $\mathcal E\in2^X$ is wrong as well. $\endgroup$ – Did Dec 18 '13 at 22:32
  • $\begingroup$ @Did: Ahhh I am blind $\endgroup$ – Prism Dec 18 '13 at 22:34

Let $E\in\mathcal{E}$.

Then $\mathcal{F}=\left\{ E\right\} $ is a countable subset of $\mathcal{E}$ and $E$ is contained in the $\sigma$-algebra generated by $\mathcal{F}$.

Then also $E\in\mathcal{N}$ (the union).

Proved is now that $\mathcal{E}\subseteq\mathcal{N}$

  • $\begingroup$ So simple! Thank you very much. I was confusing the subsets of $X$ and subsets of $2^{X}$... $\endgroup$ – Prism Dec 18 '13 at 22:39

Here's a complete proof for the reference. Taken from here. I am making this post community-wiki.

enter image description here

  • $\begingroup$ There is a typo in the beginning of third paragraph. It should say "To see that $\mathcal{M}(\mathcal{E})\subseteq\mathcal{M}$ we first notice that…" I will correct it when I get the chance to latexify the post (right now, it's only a screenshot) $\endgroup$ – Prism Dec 18 '13 at 23:20

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.