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Base- $b$ box step functions are dense in $L^{2}$ of the cube

Statement

Let $b \geq 2$ and $d \geq 1$ . In $L^{2} ([0, 1)^{d})$ for Borel Lebesgue measure, the finite linear span of indicators of base- $b$ elementary boxes is dense.

Facts & Assumptions

Given: Integers $b \geq 2$ , $d \geq 1$ , and the Borel cube with Lebesgue measure.

[F1]

Simple functions with finite-measure support are dense in $L^{p} (μ)$ for $1 \leq p < \infty$ : Every $L^{2}$ class is approximated in $L^{2}$ by a finite simple function supported on finitely many measurable sets.

[F2]

Continuity from below for measures: If $E_{N} ↑ E$ , then $λ (E_{N}) \to λ (E)$ .

Proof

technique · direct

Given: $b \geq 2$ , $d \geq 1$ , and $ε > 0$ .

1.1givenconstruct

Let $A_{b}$ be the algebra of finite unions of base- $b$ elementary boxes in $[0, 1)^{d}$ . There are countably many such boxes. For any open $O \subseteq [0, 1)^{d}$ and $x \in O$ , choose $δ_{x} > 0$ such that the relative $δ_{x}$ -ball about $x$ lies in $O$ . At every sufficiently large level $k$ , the unique level- $k$ box containing $x$ has diameter at most $\sqrt{d} b^{- k} < δ_{x}$ , hence is contained in $O$ . Therefore $O$ is the union of the countably many base- $b$ boxes contained in it. Thus $σ (A_{b})$ contains every open set and equals the Borel $σ$ -algebra.

2.1F2step 1.1algebrachoose

Let $C$ be the Borel sets $E$ for which every $η > 0$ admits $A \in A_{b}$ with $λ (E △ A) < η$ . It contains $A_{b}$ and is closed under complements and finite unions by the symmetric-difference inequalities and the algebra property. If $E_{i} \in C$ , then $E^{(N)} = ⋃_{i = 1}^{N} E_{i} ↑ E = ⋃_{i \geq 1} E_{i}$ ; [F2] lets us choose $N$ with $λ (E ∖ E^{(N)}) < η / 2$ . Approximate each of the finitely many $E_{i}$ , $i \leq N$ , within $η / (2 N)$ and union those approximants. The resulting member of $A_{b}$ is within $η$ of $E$ . Thus $C$ is a $σ$ -algebra, so step 1.1 gives $C$ equal to all Borel sets.

3.1F1step 2.1givenchoosealgebra

By [F1], choose a finite simple function $s = \sum_{j = 1}^{M} c_{j} 1_{E_{j}}$ with $∥ f - s ∥_{2} < ε / 2$ . By step 2.1 choose $A_{j} \in A_{b}$ so that $\sum_{j = 1}^{M} ∣ c_{j} ∣ λ (E_{j} △ A_{j})^{1 / 2} < ε / 2$ . The finite box step function $g = \sum_{j} c_{j} 1_{A_{j}}$ then satisfies $∥ s - g ∥_{2} \leq \sum_{j} ∣ c_{j} ∣ ∥ 1_{E_{j}} - 1_{A_{j}} ∥_{2} = \sum_{j} ∣ c_{j} ∣ λ (E_{j} △ A_{j})^{1 / 2} < ε / 2$ .

4.1F1step 3.1algebra∎

The triangle inequality gives $∥ f - g ∥_{2} \leq ∥ f - s ∥_{2} + ∥ s - g ∥_{2} < ε$ . Since $f$ and $ε$ were arbitrary, the stated finite-span density follows.

Depends on

Used by

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Sources

Art B. Owen, Monte Carlo Variance of Scrambled Net Quadrature, §6 (standard reference, not scraped)
Art B. Owen, Monte Carlo Book: The Quasi-Monte Carlo Parts, Chapter 17 §17.5 (standard reference, not scraped)

Base-bb box step functions are dense in L2L2 of the cube