Let $\nu$ denote a finite and signed measure. I am reading some notes in which the notation $$\nu\big|_{[a, b]}\ge 0$$ is used (here $a, b>0$).
Anyone could please tell me what does it mean? Clearly this notation is not defined in the notes, but I gather from the context that it could mean that for any function $f$ supported in $[a, b]$, it is $$\int_{supp(f)} f(x) d\nu(x) = \int_{supp(f)} f(x) d\nu^+(x),$$ with $\nu^+$ denotes the positive part of $\nu$.
Anyone could please provide a reference?
I would guess that this means the measure $\eta:\mathcal{B}\to (-\infty,\infty)$, given by $\eta(A)=\nu(A\cap [a,b])$ for all $A\in \mathcal{B}$ measurable.
If you care about what measures you can plug into measure, it could also mean restricting to the natural $\sigma$-sub-algebra $\mathcal{B'}=\{ C\subseteq [a,b]: C=A\cap [a,b],\; A\in \mathcal{B} \}$, and defining the measure as above.
Perhaps Restriction of a finite measure to a set or Correct notation for restriction of measure can help you.
Ultimately it depends on the intention of who wrote the notes, so it could be something different.