Outer Measure of the Unit Interval

Today, we’re doing two for one since they’re both fairly straightforward.

2.2.5 - Real Analysis: Royden

By using properties of outer measure, prove that the interval $[0, 1]$ is not countable.

Proof:

The outer measure of $[0, 1]$ is 1. Suppose that $[0, 1]$ is countable. We have that the outer measure of any countable set 0, and thus that contradicts $m^*([0, 1]) = 1$, thus $[0, 1]$ is not countable.

2.2.6 - Real Analysis: Royden

Let $A$ be the set of irrational numbers in the interval $[0, 1]$. Prove that $m^*(A) = 1$.

Proof:

First, note that

\[A = [0, 1] \setminus (\mathbb{Q} \cap [0, 1])\]

Since $\mathbb{Q}$ is countable, we have that $m^*(\mathbb{Q} \cap [0, 1]) = 0$. Hence,

\[\begin{aligned} m^*([0, 1]) &= m^*(A) + m^*(\mathbb{Q} \cap [0, 1]) \\ 1 &= m^*(A) \end{aligned}\]