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The standard definition of a measurable space is as a tuple of a set and an associated $\sigma$-algebra, say, $\left(\Omega, \Sigma \right)$.

Given a $\sigma$-algebra, $\Sigma$, we could extract the original "universe" set, $\Omega$, by taking the complement of the empty set and then pair them together to create a measurable space, $\left(\Omega, \Sigma \right)$. And given a measurable space, $\left(\Omega, \Sigma \right)$, we could get the $\sigma$-algebra, $\Sigma$, by "forgetting" the universe set.

So there is an (almost trivial) isomorphism between measurable spaces and $\sigma$-algebras.

Is this true? If so, why do we make a distinction between the two objects when distinctions between isomorphic things are so often elided in mathematics?

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  • $\begingroup$ If you ask me to take the complement of the empty set I wll ask back complement with what ? When I don't have $\Omega$ to perform this I will have no idea what to do. $\endgroup$
    – Kurt G.
    Commented Apr 9, 2022 at 2:47
  • $\begingroup$ You are correct. I think the main reason for doing this is that in most cases one considers the set $X$ to be more relevant than the sigma algebra, since there is often a "natural" sigma algebra attached to a space (e.g., the Bored sigma algebra on $\Bbb{R}^d$). If one would define a measurable space as just the sigma algebra, then this approach of (later) just considering a set and maybe not even mentioning the sigma algebra would be somewhat awkward. Also note that a group is determined by the domain of definition of the multiplication. Still one mentions the group. $\endgroup$
    – PhoemueX
    Commented Apr 9, 2022 at 7:36
  • $\begingroup$ @KurtG. So the $\sigma$-algebra is just a set? Not a set with equipped with operations of union, intersection and complement, like a semigroup is a set equipped with an associative operation? I guess I've never seen anyone define the laws that union, intersection and complement would have, so they must be talking about the usual set operations. This makes sense -- I think the names "$\sigma$-algebra" and "$\sigma$-field" lead me to think about it as something like, well, an algebra or a field. That is, as a set equipped with certain operations. $\endgroup$ Commented Apr 9, 2022 at 14:00
  • $\begingroup$ A $\sigma$-algebra is a family of sets (subsets of $\Omega$). It is closed under complement and countable union. Hence closed under countable intersection as well. I do not belong to those who think that the notions of complement, union and intersection need further definitions. Somewhere one has to start and get going. $\endgroup$
    – Kurt G.
    Commented Apr 9, 2022 at 18:49
  • $\begingroup$ @KurtG Would you care to convert your comment into an answer so that I can accept it and close this question? $\endgroup$ Commented Apr 11, 2022 at 15:36

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