What is a set mapping? Short question, what is a set mapping? I find it difficult using a set as an argument. I am struggling with measure theory at the moment and the example is a function
$f: S \to T$ ,
s.t.
$f^{-1}(B) = \{s \in S; f(s) \in B \}$ with $B \subset T$
If someone could explain this.
Thanks.

Let it be that $S$ and $T$ are sets and that $f:S\to T$ is a function.
We have the powersets $\mathcal{P}(S)$ and $\mathcal{P}(T)$.
Function $f$ induces/determines a function $\mathcal{P}(S)\to\mathcal{P}(T)$ that is prescribed by: $$A\mapsto\{f(x)\mid x\in A\}$$ The output is abbreviated by $f(A)$ so we have: $$A\mapsto f(A)$$
Function $f$ induces/determines a function $\mathcal{P}(T)\to\mathcal{P}(S)$ that is prescribed by: $$B\mapsto\{x\in S\mid f(x)\in B\}$$ The output is abbreviated by $f^{-1}(B)$ so we have: $$B\mapsto f^{-1}(B)$$
Caution: we are not dealing with any inverse map of $f$ as the notation seems to suggest.
