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.

A set mapping is a map between sets. This carries remarkably little information on its own.
A function $f: S \longrightarrow T$ is a way of associating to each $s$ in $S$ some $t$ in $T$. The $t$ associated to an $s$ is usually denoted by $f(s)$.
Alternately, a function $f: S \longrightarrow T$ is a particular relation, i.e. a set of ordered pairs $(s, t)$ with $s$ in $S$ and $t$ in $T$, with the property that there is exactly one ordered pair containing $s$ for each $s$ in $S$.
In your question, you've mentioned $f^{-1} B$, which might be read as "the preimage of $B$". You have defined this as those elements in $S$ that are sent to elements in $B$ by the function $f$ (or rather, elements in $S$ associated to elements in $B$ under the relation given by $f$).
For instance, suppose $g: \mathbb{Z} \longrightarrow \mathbb{Z}$ is the map
$$ g(n) = \begin{cases} 0 & n \text{ is even} \\
1 & n \text{ is odd} \end{cases}$$
Then you might ask what is $g^{-1} 0$? And it is exactly the even integers.
