Answer to Question #310731 in Real Analysis for ABC2468

Question

Prove that a subset of a set of measure zero has measure zero.

Answer 1

By the properties of the measure $\mu$ :

$A\subset B\Rightarrow \mu A\leq\mu B$

But $\mu C\geq0$ for any measurable set $C$. So, if the measure of the set $B$ is zero:

$0\leq\mu A\leq\mu B=0\Rightarrow\mu A=0$