Exercise 1.2.3 (The outer measure axioms)

Notice that by non-negativity of the elementary measure, the Lebesgue outer measure has to remain non-negative as well.

(i)

$\varnothing$ is contained in any degenerate box ${\{a\}}^d$ of elementary measure 0; thus, $m^{\text{∗}}(\varnothing )\le 0$. By the previous remark, Lebesgue outer measure is non-negative; thus, it must be zero.

(ii)

Let $E\subseteq F\subseteq {ℝ}^d$. Since any outer cover of $F$ is also an outer cover of $E$, we have an obvious relation $$\left\{\underset{n=1}{\overset{\infty }{\sum }}|B_n|:\begin{array}{c}{B}_n\text{ are boxes}\hfill \\ E\subset \underset{n=1}{\overset{\infty }{\bigcup }}{B}_n\hfill \end{array}\right\}\supseteq \left\{\underset{n=1}{\overset{\infty }{\sum }}|C_n|:\begin{array}{c}{C}_n\text{ are boxes}\hfill \\ F\subset \underset{n=1}{\overset{\infty }{\bigcup }}{C}_n\hfill \end{array}\right\}.$$

The infimum of a superset is always smaller than the infimum of any of its subsets; thus,

$$m^{\text{∗}}(E)\le m^{\text{∗}}(F).$$

(iii)

Let $E_1,E_2,\dots \subset {ℝ}^d$ be a countable sequence of sets; our goal is to demonstrate that $$\inf <!--\; FUNCTION\; APPLICATION\; -->\left\{\underset{n=1}{\overset{\infty }{\sum }}|B_n|:\begin{array}{c}{B}_n\text{ are boxes}\hfill \\ \underset{i=1}{\overset{n}{\bigcup }}{E}_i\subset \underset{n=1}{\overset{\infty }{\bigcup }}{B}_n\hfill \end{array}\right\}\le \underset{i=1}{\overset{n}{\sum }}\inf <!--\; FUNCTION\; APPLICATION\; -->\left\{\underset{n=1}{\overset{\infty }{\sum }}|\underset{n}{\overset{(i)}{C}}|:\begin{array}{c}\underset{n}{\overset{(i)}{C}}\text{ are boxes}\hfill \\ E_i\subset \underset{n=1}{\overset{\infty }{\bigcup }}\underset{n}{\overset{(i)}{C}}\hfill \end{array}\right\}(+𝜖)$$

for every $𝜖>0$. By definition of infimum, for $𝜖>0$ we can find a cover $C_1^{(i)},\dots$ of $E_i$ such that ${\sum <!--\; FUNCTION\; APPLICATION\; -->}_{n=1}^{\infty }|C_n^{(i)}|-m^{\text{∗}}(E_i)\le 𝜖∕2^i$. Thus, by the countable axiom of choice, we can construct a double infinite sequence

$${(C_{(i,n)})}_{i\in \mathbb{N},n\in \mathbb{N}}\text{ such that }\forall <!--\; FUNCTION\; APPLICATION\; -->i\in \mathbb{N}:E_i\subseteq \underset{n=1}{\overset{\infty }{\bigcup }}{C}_{(i,n)}\text{ and }\underset{n=1}{\overset{\infty }{\sum }}|C_{(i,n)}|\le m^{\text{∗}}(E_i)+𝜖∕2^i.$$

But we immediately notice that (cf. Tonelli’s theorem for series)

$$\underset{i\in \mathbb{N}}{\bigcup }{E}_i\subseteq \underset{i\in \mathbb{N}}{\bigcup }\underset{n\in \mathbb{N}}{\bigcup }{C}_{(i,n)}=\underset{(i,n)\in {\mathbb{N}}^2}{\bigcup }{C}_{(i,n)}⟹\underset{(i,n)\in {\mathbb{N}}^2}{\sum }|C_{(i,n)}|\in \left\{\underset{n=1}{\overset{\infty }{\sum }}|B_n|:\begin{array}{c}{B}_n\text{ are boxes}\hfill \\ \underset{i=1}{\overset{n}{\bigcup }}{E}_i\subset \underset{n=1}{\overset{\infty }{\bigcup }}{B}_n\hfill \end{array}\right\}$$

which satisfies

$$\begin{array}{lll}\hfill m^{\text{∗}}\left(\underset{i=1}{\overset{n}{\bigcup }}{E}_i\right)\le \underset{(i,n)\in {\mathbb{N}}^2}{\sum }|C_{(i,n)}|=\underset{i\in \mathbb{N}}{\sum }\underset{n\in \mathbb{N}}{\sum }|C_{(i,n)}|\le \underset{i\in \mathbb{N}}{\sum }\left(m^{\text{∗}}(E_i)+𝜖∕2^i\right)=\underset{i\in \mathbb{N}}{\sum }{m}^{\text{∗}}(E_i)+𝜖& & \hfill \end{array}$$

as desired.

First, $\varnothing$ is a box of volume 0 ; another proof is by noting that $\varnothing \subset {\left[0,\frac{1}{n}\right]}^d$ for all $n$, so $m^{\text{∗}}(\varnothing )\le \underset{n}{\inf <!--\; FUNCTION\; APPLICATION\; -->}m\left({\left[0,\frac{1}{n}\right]}^d\right)=0.$

For $A\subset C$ we have that the infimum used to obtain $m^{\text{∗}}(A)$ is over a larger set that the one used to obtain the infimum for $m^{\text{∗}}(C)$.

Let ${\left(A_n\right)}_n$ be a sequence of sets. Fix $𝜀>0.$ For every $n$ let ${\left(B_{n,k}\right)}_k$ be a sequence of boxes such that $A_n\subset {\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_k{B}_{n,k}$ and

Consider the sequence of boxes ${\left(B_{n,k}\right)}_{n,k}$. We have that

and so

Taking $𝜀\to 0$ we have countable subadditivity.