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 $R^{d}$

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

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

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

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

15.07.2020

Lebesgue outer measure. Finite additivity (for separated sets)

Separated sets in Rd

Separated sets in $R^{d}$