Lectures on Functional Analysis and the Lebesgue Integral by Vilmos Komornik

Author:Vilmos Komornik
Language: eng
Format: epub, pdf
Publisher: Springer London, London

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Using Proposition 6.2 we may investigate the density of sets:

Definition

A measurable set A set has density d at a point if

(6.4)

for every sequence (I n ) of non-degenerate intervals, containing x and satisfying .

We always have 0 ≤ d ≤ 1; for example a set has density one at each point of its interior. Much more is true:

Proposition 6.4 (Lebesgue)

8 Every measurable set A set has density one at a.e. point of A.

Proof

Since density is a local property, we may assume that A is bounded. Then χ A integrable, and its indefinite integral F satisfies F′ = χ A a.e. by Proposition 6.2 (p. 200).

The equality F′(x) = χ A (x) means that (6.4) holds with d = χ A (x) if x is an endpoint of each interval I n . The general case follows from the identity

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