Let $\left(X,\mathscr{A},\mu\right)$ be a measure space. In René Schilling's Measures, Integrals and Martingales, Definition 11.1 reads:
A $\left(\mu-\right)$null set $N\in\mathscr{N}_\mu$ is a measurable set $N\in\mathscr{A}$ satisfying $$ N \in \mathscr{N}_{\mu} \Longleftrightarrow N \in \mathscr{A} \quad \text { and } \quad \mu(N)=0. $$
There are two things that confuses me about this definition:
- $\mathscr{N}_\mu$ has not been defined.
- In the first line of the definition we picked an $N$ that is in both $\mathscr{N}_\mu$ and $\mathscr{A}$, so what is the purpose of saying that it must satisfy $N \in \mathscr{N}_{\mu} \Longleftrightarrow N \in \mathscr{A}$?