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Finite linear combinations of box indicators are dense in $L^{p} (R^{n})$ for $1 \leq p < \infty$

Statement

Assume the Axiom of Countable Choice.

Let $1 \leq p < \infty$ . Finite linear combinations of indicator functions of boxes are dense in $L^{p} (R^{n})$ .

Facts & Assumptions

Given: The Axiom of Countable Choice, $1 \leq p < \infty$ , $ε > 0$ , and $f \in L^{p} (R^{n})$ .

[L1]

Simple functions with finite-measure support are dense in $L^{p}$ (Simple functions with finite-measure support are dense in $L^{p} (μ)$ for $1 \leq p < \infty$ ).

[L2]

Every finite-measure measurable set can be approximated in symmetric difference by a finite union of boxes (A finite-measure measurable set in $R^{n}$ is approximable in measure by a finite union of boxes).

[L3]

Minkowski's inequality is available in $L^{p}$ (Minkowski's inequality for integrals, including $p = \infty$ ).

Proof

technique · direct

1.1L1L2givenchoose

By [L1], choose a simple function [L1, L2, given, choose] $s = \sum_{j = 1}^{m} a_{j} 1_{E_{j}}$ with each $E_{j}$ of finite measure and $∥ f - s ∥_{p} < ε / 2$ . For each $j$ , choose a finite union of boxes $B_{j}$ with $λ_{n} (E_{j} △ B_{j}) < {(\frac{ε}{2 m (1 + \sum_{j = 1}^{m} ∣ a_{j} ∣)})}^{p}$ by [L2].

2.1L3step 1.1algebra

Put [L3, step 1.1, algebra] $t : = \sum_{j = 1}^{m} a_{j} 1_{B_{j}} .$ Then $∥ s - t ∥_{p} \leq \sum_{j = 1}^{m} ∣ a_{j} ∣ ∥ 1_{E_{j}} - 1_{B_{j}} ∥_{p} = \sum_{j = 1}^{m} ∣ a_{j} ∣ λ_{n} (E_{j} △ B_{j})^{1 / p} < ε / 2$ by [L3] and the choice of the $B_{j}$ .

3.1step 1.1step 2.1algebra∎

Therefore [step 1.1, step 2.1, algebra] $∥ f - t ∥_{p} \leq ∥ f - s ∥_{p} + ∥ s - t ∥_{p} < ε .$ Since $t$ is a finite linear combination of box indicators, these functions are dense in $L^{p} (R^{n})$ .

Finite linear combinations of box indicators are dense in $L^{p} (R^{n})$ for $1 \leq p < \infty$

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

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

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

Finite linear combinations of box indicators are dense in $L^{p} (R^{n})$ for $1 \leq p < \infty$