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A finite-measure measurable set in $R^{n}$ is approximable in measure by a finite union of boxes

Statement

Assume the Axiom of Countable Choice.

Let $E \subseteq R^{n}$ be Lebesgue measurable with $λ_{n} (E) < \infty$ . For every $ε > 0$ there is a finite union of boxes $B$ such that

λ_{n} (E △ B) < ε .

Facts & Assumptions

Given: The Axiom of Countable Choice, a natural number $n \geq 1$ , a Lebesgue measurable set $E \subseteq R^{n}$ with finite measure, and $ε > 0$ .

[L1]

Outer regularity gives an open set $U \supseteq E$ with $λ_{n} (U ∖ E) < ε / 2$ (Assuming countable choice, the Lebesgue outer measure of an arbitrary subset of $R^{n}$ is the infimum of the measures of the open sets containing it).

[L2]

Every open subset of $R^{n}$ is a countable disjoint union of dyadic cubes (Every open subset of $R^{n}$ is the union of a countable pairwise disjoint family of dyadic cubes).

[L3]

Continuity from below applies to increasing unions of measurable sets (Continuity from below for measures).

[L4]

Measure is monotone and $λ_{n} (U ∖ F) = λ_{n} (U) - λ_{n} (F)$ when $F \subseteq U$ and $λ_{n} (U) < \infty$ (Measures are monotone, Measure of a set difference when the smaller set has finite measure).

Proof

technique · direct

1.1

By [L1], choose an open set $U \supseteq E$ with [L1, L4, given, choose] $λ_{n} (U ∖ E) < ε / 2$ . Because $λ_{n} (E) < \infty$ , monotonicity gives $λ_{n} (U) \leq λ_{n} (E) + λ_{n} (U ∖ E) < \infty .$

L1L4givenchoose

1.2

Write $U = ⨆_{k \geq 1} Q_{k}$ as a countable pairwise disjoint union [L2, L3, L4, choose] of dyadic cubes by [L2], and set $B_{m} : = ⋃_{k = 1}^{m} Q_{k}$ . Then $B_{m} ↑ U$ , so [L3] gives $λ_{n} (B_{m}) \to λ_{n} (U)$ . Hence for some $m$ , $λ_{n} (U ∖ B_{m}) = λ_{n} (U) - λ_{n} (B_{m}) < ε / 2.$

L2L3L4choose

2.1

Put $B : = B_{m}$ . Since $E \subseteq U$ and $B \subseteq U$ , [step 1.1, step 1.2, L4, algebra] $E △ B \subseteq (U ∖ E) \cup (U ∖ B),$ so $λ_{n} (E △ B) \leq λ_{n} (U ∖ E) + λ_{n} (U ∖ B) < ε .$ The set $B$ is a finite union of boxes because each dyadic cube is a box.

step 1.1step 1.2L4algebra∎

Depends on

Used by

Finite linear combinations of box indicators are dense in Lᵖ(ℝⁿ) for 1 ≤ p < ∞ Theorem

Dependency tree · two levels

41 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Terence Tao, An Introduction to Measure Theory (standard reference, not scraped)
Walter Rudin, Real and Complex Analysis, 3rd ed. (standard reference, not scraped)

A finite-measure measurable set in RnRn is approximable in measure by a finite union of boxes