Prove the triangle inequality for Lebesgue outer measure

Answer 1

Let $A,B$ be subsets of $\mathbb{R}^n$, and let $E$ be an arbitrary set. We want to prove that
$$
m^*(A) + m^*(B) \geq m^*(A\cup B),
$$
where $m^*$ is the Lebesgue outer measure.

First, we note that $m^*(E\cup F) \leq m^*(E) + m^*(F)$ for any sets $E$ and $F$. This is because any cover of $E\cup F$ can be split into covers of $E$ and $F$, so taking the infimum over all covers of $E$ and $F$ gives the desired inequality.

Using this result, we have
\begin{align*}
m^*(A\cup B) &= m^*((A\setminus B)\cup(B\setminus A)\cup(A\cap B)) \\
&\leq m^*(A\setminus B) + m^*(B\setminus A) + m^*(A\cap B) \\
&\leq m^*(A) + m^*(B),
\end{align*}
where the second inequality follows from the fact that $A\setminus B$ and $B\setminus A$ are disjoint, and hence their covers can be combined. This completes the proof of the triangle inequality for Lebesgue outer measure.