Consider two measurable spaces: first $\mathbb Z$ with the usual sigma algebra, and second $\mathbb R$ with the sigma algebra generated by $\{[n, n+1): n \in \mathbb Z\}$. The sigma algebras are equivalent but there is no bijection between the underlying sets.
Informally, it seems like these two spaces should be equivalent from the perspective of probability theory: any probability measure or random variable measurable on one of them has an equivalent on the other.
Is there a notion of equivalence between measurable spaces that makes these two spaces equivalent? Alternatively, is there an example of two measurable spaces with equivalent sigma algebras, that are meaningfully different from the perspective of probability theory?