Questions about null sets

Question

I'm working in an arbitrary measure space, denoted by $<X,M,\mu>$, where X is a non-empty set, $M$ is a sigma-algebra and $\mu$ is the measure, $\mu:M\rightarrow[0,\infty]$. Suppose $N$ is a null set, ie, $\mu(N)=0$, my questions are:

1. Why does $N$ belongs to $M$? Is it because $\mu(N)$ is defined?

2. Suppose $E \subset N$, where $N \in M$ and $\mu(N)=0$, why $E \in M$?

Answer 1

1. Yes, if $\mu(A)$ is defined that means that $A\in M$, since $M$ is the domain of the function $\mu$.

2. This isn't necessarily true. For example, let $M$ be the sigma-algebra of Borel sets on $\Bbb R$ and $\mu$ Lebesgue measure. Then $|M|=2^{\aleph_0}$. The Cantor set $C$ has measure zero and $|C|=2^{\aleph_0}$, so that $C$ has $2^{2^{\aleph_0}}$ subsets. So it has non-Borel subsets: there exists $N\subset C$ with $N\notin M$.

A measure space with the property that subsets of measure zero sets are always measurable is called a complete measure space.