Q:

Prove that the restriction of a Lebesgue measure on a subset is a complete measure

Given a Lebesgue measure $m$ on a $\sigma$-field $\mathcal M$ and Lebesgue measurable set $B$ let 
$ \mathcal M_B = \{A \cap B\colon A \in \mathcal M\}$ and
$ m_B(A) = m(A) $
How do I first show that $\mathcal M_B$ is a $\sigma$-field on subsets of $B$.

$B$ belongs to $\mathcal M_B$ since $B = B \cap B $ and $B \in \mathcal M$
$ \bigcup_{i}(A_i \cap B) = (\bigcup_iA_i)\cap B $
How do I prove that if $A \cap B \in \mathcal M_B$, then $ (A \cap B) ^c \in \mathcal M_B$?

A:

Take the solution to this problem : Show: $\mathcal{G}:=\left\{B\in\mathcal{B}(\mathbb{R}^n)|t+B\in\mathcal{B}(\mathbb{R}^n)\right\}$ is $\sigma$-Algebra, and apply it to the inclusion function from $B$ to $\mathbb{R}$

