Why distinguish between a $\sigma$-algebra and a measurable space?

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?

• 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. Commented Apr 9, 2022 at 2:47
• 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. Commented Apr 9, 2022 at 7:36
• @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. Commented Apr 9, 2022 at 14:00
• 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. Commented Apr 9, 2022 at 18:49
• @KurtG Would you care to convert your comment into an answer so that I can accept it and close this question? Commented Apr 11, 2022 at 15:36