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?