I have a very (very) basic question about the composition of "functions". Take $f \in L^1(0,1)$ and a continuous function $g:\mathbb{R} \rightarrow \mathbb{R}$. I simply wonder what is meant by an inequality like $$\tag{1} g \circ f \leq 0 \text{ almost everywhere in } (0,1).$$ $g$ is a function but $f$ is an equivalence class so a priori $g \circ f$ doesn't make sense.
In (1), is $g \circ f$ understood in fact as an equivalence class, defined by $$g \circ f:=\{g \circ h\ | \ h \text{ belongs to the class of } f\}?$$
If so, is it correct to say that (1) then means that there exists a measurable subset $N \subset (0,1)$ with measure zero such that, for all $x \in (0,1) \backslash N$, we have $g(h(x)) \leq 0$ for some/all the representatives $h$ of $f$ ? In particular, $N$ does not depend on the representative of $f$ ?
This means that the equivalence class of $g \circ f_0$ doesn't depend on the representative $f_0$ of the class $f$ : for all $f_0$ and $f^\prime_0$ equal almost everywhere, then $g\circ f_0 = g \circ f^\prime_0$ almost everywhere. And for all representative $h_0$ of the class $g \circ f$, there exists a measure zero $N=N(h_0)$ such that $\forall x \in N^c, \, h_0(x) \leqslant 0$