Math and science::Analysis::Tao, measure::02. Lebesgue measure

# Lebesgue outer measure. Finite additivity (for separated sets)

For Lebesgue outer measure, we left Pallet Town, so to speak, with just three properties: the measure of an empty set is zero, monotonicity and [something]. Our next step is additivity, though a restricted version of additivity: restricted to the union of just two sets, and the sets must be [something].

#### Separated sets in $$\mathbb{R}^d$$

Let $$E, F \subset \mathbb{R}^d$$ be such that $$\operatorname{dist}(E, F) > 0$$, where $$\operatorname{dist}(E, F)$$ is the distance between $$E$$ and $$F$$, defined as:

$\operatorname{dist}(E, F) := \inf(\{ |x - y| : x \in E, y \in F \} )$

When $$\operatorname{dist}(E,F) > 0$$, $$E$$ and $$F$$ are said to be separated sets.

Let $$E, F \subset \mathbb{R}^d$$ be separated sets ($$\operatorname{dist}(E, F) > 0$$). Then [what can we say about the Lebesgue outer measure?].