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?].