Consider $\mathbb{P}, \mu$ measures on the measurable space $(\Omega, \mathcal{F})$. Suppose $\mathbb{P}$ has density $p$ with respect to $\mu$. Let $A \in \mathcal{F}$.
Statement: $\mathbb{P}(A)=0\rightarrow p(\omega)=0$ for $\mu-$almost every $\omega \in A$ (from van der Vaart, Asymptotic Statistics, p.86).
Question: What does it mean "for $\mu-$almost every $\omega \in A$"?
My attempt: $\mathbb{P}(A)=0 \leftrightarrow \int_{A}d\mathbb{P} =\int_{A}p(\omega )d\mu=0\rightarrow p(\omega)=0 \forall \omega \in A$ provided that $\mu(A)>0$.
The meaning of "for $\mu-$almost every $\omega \in A$" is mine "$...\forall \omega \in A$ provided that $\mu(A)>0$"?