Table of Contents

Example Problems Regarding the Lebesgue Outer Measure
So far we have shown the following properties regarding the Lebesgue outer measure:
 1. For any finite or countably infinite subset $E$ of $\mathbb{R}$:
 $m^*$ has the monotonicity property. If $A \subseteq B$ then:
 3. If $I$ is any interval then:
 4. $m^*$ is translation invariant. For any subset $E$ of $\mathbb{R}$ and for any $a \in \mathbb{R}$ we have that:
 5. $m^*$ is countably subadditive. For any sequence $(A_n)_{k=1}^{\infty}$ of subsets of $\mathbb{R}$ we have that:
We now look at some example problems involving the Lebesgue outer measure.
Example 1
Prove that $[0, 1]$ is an uncountable subset of $\mathbb{R}$.
If $[0, 1]$ were countable then $m^*([0, 1]) = 0$. But $[0, 1]$ is an interval and so $m^*([0, 1]) = l([0, 1]) = 1$. Therefore $[0, 1]$ must be an uncountable subset of $\mathbb{R}$.
Example 2
Let $I$ be the set of all irrational numbers contained in the interval $[0, 1]$. Prove that $m^*(I) = 1$.
The set of all rational numbers contained in $[0, 1]$ is a countable set and so by countable subadditivity we have that:
(6)But since $I \subseteq [0, 1]$ we have the monotonicity of the Lebesgue outer measure that:
(7)Therefore we conclude that $m^*(I) = 1$.
Example 3
Let $A, B \subseteq \mathbb{R}$. Prove that if $m^*(A) = 0$ then $m^*(A \cup B) = m^*(B)$.
Since $B \subseteq A \cup B$ we have by the monotonicity property of the Lebesgue outer measure that:
(8)Furthermore by the countable subadditivity of the Lebesgue outer measure we have that:
(9)Therefore we conclude that $m^*(A \cup B) = m^*(B)$.