In Probability and Stochastics, (Cinlar), isomorphic measurable spaces are defined as follows:
So we have a bijection $f$. But then it's emphasized that we have a functional inverse $\hat{f}$. Why is this distinction necessary? Is $\hat{f}$ not also a bijection? Are there other inverse functions (or I guess, are there multiple bijective maps from $(F, \mathcal{F})$ to $(E, \mathcal{E})$)?
I know isomorphism means that there's a bijection that preserves structure (e.g., for vector spaces, linearity is preserved), it's not clear to me what structure exactly is being preserved (since both spaces are already measurable)?