Intuition behind this inverse image notation on measure theory.

Question

So my professor for measure theory when talking about $\sigma$-algebras says statements like:

$$ \{f \in \tilde{E}\} \in \mathcal{F} $$

when he wants to denote $f^{-1}(\tilde{E}) \in \mathcal{F}$.

Is this a commonly accepted notation in measure theory/stochastics? If it is, what is the intuition behind this notation? I don't at all see how saying $f \in \tilde{E}$ conjures notions of inverse images. If anything to me it seems like notation for an actual image.

Thanks for any help!

Answer 1

In this context, you can think of $\{ f \in \tilde E \}$ as a shorthand version of $\{ x \mid f(x) \in \tilde E \}$.