Exercise 1.4.27 (Lebesgue measure space is completion of Boolean measure space)

By Exercise 1.4.26 the completion of $B[{ℝ}^d]$ is given by

where is a collection of sub-null sets of the Borel measure space. We demonstrate that $\overline{B}[{ℝ}^d]=L[{ℝ}^d]$.

• Let $\overline{E}\in \overline{B}$, i.e., $E=B\cup N$ for a Borel measurable $B\in B$ and a Borel sub-null set $N\subseteq N^{\prime }\in N$. The former is contained in $L[{ℝ}^d]$ since every Borel measurable set is Lebesgue measurable (p. 72). The latter is a Lebesgue null set, since by the monotonicity of the Lebesgue outer measure we have $m^{\text{∗}}(N)\le m(N)=0$. Thus, their union is a Lebesgue measurable set.
• Let $E\in L[{ℝ}^d]$. By Exercise 1.2.19 $E$ is a $G_{\delta }$ set with a Lebesgue null set removed, i.e., $E={\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_{n=1}^{\infty }{U}_{n}N$ for open sets ${(U_n)}_{n\in \mathbb{N}}$ and a Lebesgue null set $N$. The former intersection is a Borel measurable set, and is thus contained in $B[{ℝ}^d]$. The latter is a Borel sub-null set. This is because it is a Lebesgue null set, i.e., by Exercise 1.2.19 for each $n\in \mathbb{N}$ we can find an open set $V_n$ such that $N\subseteq V_n$ and $m(V_n)=m(V_n∕N)\le 1∕n$. Then, ${\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_{n=1}^{\infty }{V}_n$ is a Borel measurable set, and it is in particular a Borel null set. But we have $N\subseteq {\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_{n=1}^{\infty }{V}_n$ and thus $N$ is a Borel sub-null set. Thus the union of both sets is contained in $\overline{B}$.