Exercise 1.4.11 (Lebesgue $\sigma$-algebra)

Since we have already proven these sets to be Boolean algebras in Exercise 1.4.1 and Examples 1.4.4 - 1.4.6, it is sufficient to only upgrade the finite additivity to the countably infinite additivity.

(i)

(Elementary Boolean algebra) Let $[n,n+1{\text{)}}_{n=0}^{\infty }$ be an countably infinite collection of boxes in $ℝ$. Then, their union is ${\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_{n=0}^{\infty }[n,n+1)=[0,+\infty )$, and the latter is not representable as a finite union of boxes (any box is bounded, and so is their finite union).

(ii)

(Jordan Boolean algebra) Consider the bullet set, i.e., the set of rational numbers ${(q)}_{q\in \mathbb{Q}\cap [0,1]}$ in the unit interval. Each of these points is Jordan null sets Exercise 1.1.12, but their countable union ${\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_{q\in \mathbb{Q}\cap [0,1]}q=\mathbb{Q}\cap [0,1]$ is not by Exercise 1.1.18.

(iii)

(Null $\sigma$-algebra) Let ${\left(A\right)}_{n\in \mathbb{N}}$ be an arbitrary collection of Lebesgue null sets in ${ℝ}^d$. Then, by countable subadditivity of the Lebesgue outer measure $m^{\text{∗}}({\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_{n=1}^{\infty }{A}_n)\le {\sum }_{n=1}^{\infty }{m}^{\text{∗}}(A_n)=0$. Thus, ${\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_{n=1}^{\infty }{A}_n$ is a null set as well.

(iv)

(Lebesgue $\sigma$-algebra) Let ${\left(E\right)}_{n\in \mathbb{N}}$ be an arbitrary collection of Lebesgue measurable sets in ${ℝ}^d$. Then, their union ${\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_{n=1}^{\infty }{E}_n$ is Lebesgue measurable as well by Lemma 1.2.13 (vi).