= Completeness of Lebesgue measure
Every <subset> $B$ of a <Lebesgue null set> is a <Lebesgue measurable set> and is itself a <null set>. In fact the same conclusion follows directly from $m^*(B)=0$, without assuming measurability in advance. For any $A\subseteq\mathbb R$, <monotonicity of Lebesgue outer measure> gives $m^*(A\cap B)=0$ and $m^*(A\setminus B)\leq m^*(A)$. Therefore
$$
m^*(A\cap B)+m^*(A\setminus B)\leq m^*(A).
$$
The reverse inequality follows from <countable subadditivity of Lebesgue outer measure>, applied to the union $A=(A\cap B)\cup(A\setminus B)$. This is <Carathéodory's criterion>, proving measurability. If $B$ is a subset of an outer-null $N$, its outer measure is zero by the same monotonicity. Thus the restriction of outer measure to measurable sets is a <complete measure>.
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