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Continuous functions are dense in Lp for compactly supported finite measures

Statement

Let $μ$ be a finite regular Borel measure on $R^{d}$ concentrated on a compact set $K \subseteq R^{d}$ , and let $1 \leq p < \infty$ . Form the real $L^{p} (μ)$ space using the Borel $σ$ -algebra. Then restrictions to $K$ of real-valued compactly supported continuous functions on $R^{d}$ are dense in $L^{p} (μ; R)$ .

Statement references

Leshno et al., proof of Theorem 1, Lp approximation step

Facts & Assumptions

Given: A finite regular Borel measure $μ$ concentrated on compact $K$ , an exponent $1 \leq p < \infty$ , a real-valued Borel class $[f] \in L^{p} (μ; R)$ , and $ε > 0$ .

[F1]

$L^{p} (μ)$ consists of measurable functions with finite ${(\int ∣ f ∣^{p} d μ)}^{1 / p}$ , with functions identified when equal $μ$ -almost everywhere (The function space $L^{p} (μ)$ for $0 < p < \infty$ , The space $L^{p} (μ)$ as the quotient by null functions).

[F2]

A measure is a nonnegative countably additive set function on a sigma-algebra, and regularity here means Borel sets can be approximated from inside by compact sets and outside by open sets with arbitrarily small measure excess (Measures on sigma-algebras); regularity is the hypothesis on $μ$ .

[F3]

If $F$ is compact and $O$ is open with $F \subset O$ , there is a smooth compactly supported cutoff $ψ$ with $0 \leq ψ \leq 1$ , $ψ = 1$ near $F$ , and $supp ψ \subset O$ (Test function cutoffs and euclidean localization).

[F4]

Dominated convergence applies to $∣ f - f_{M} ∣^{p} \leq ∣ f ∣^{p}$ and $f_{M} \to f$ almost everywhere (Dominated convergence).

[F5]

For $1 \leq p < \infty$ , the quotient $L^{p} (μ)$ has a norm whose triangle inequality is Minkowski's inequality (The $L^{p}$ norm descends to the quotient and makes $L^{p}$ a normed space for $1 \leq p \leq \infty$ ).

Proof

1.1givenF1F4F5choose

Choose a measurable representative $f$ of $[f]$ . Truncate it to $f_{M} = \max (- M, \min (f, M))$ . Since $∣ f - f_{M} ∣^{p} \leq ∣ f ∣^{p}$ and $f_{M} \to f$ pointwise wherever $f$ is finite, [F4] makes $∥ f - f_{M} ∥_{L^{p} (μ)} < ε / 2$ for some large $M$ . Partition $[- M, M]$ into finitely many intervals of length at most $δ$ and replace each value of $f_{M}$ by a fixed endpoint of its interval. If $μ (K) > 0$ , choose $δ$ with $δ μ (K)^{1 / p} < ε / 2$ ; if $μ (K) = 0$ , any $δ > 0$ suffices. The resulting finite-valued Borel simple function $s$ satisfies $∣ s - f_{M} ∣ \leq δ$ pointwise, so $∥ s - f_{M} ∥_{L^{p} (μ)} \leq δ μ (K)^{1 / p}$ . By [F5] and the triangle inequality, $∥ s - f ∥_{L^{p} (μ)} < ε$ . Since $μ$ is concentrated on $K$ , replace $s$ by $1_{K} s$ : this finite Borel simple function represents the same class and admits a representation with every set $E_{j} \subseteq K$ . Thus such supported simple functions are dense in $L^{p} (μ)$ .

2.1step 1.1F2F3choose

Write such a simple function as $s = \sum_{j = 1}^{N} c_{j} 1_{E_{j}}$ with Borel $E_{j} \subseteq K$ . For any $η > 0$ , regularity [F2] gives compact $F_{j} \subseteq E_{j}$ and open $O_{j} \supseteq E_{j}$ with $μ (O_{j} ∖ F_{j}) < η$ . By [F3] choose $ψ_{j} \in C_{c}^{\infty} (O_{j})$ with $0 \leq ψ_{j} \leq 1$ and $ψ_{j} = 1$ on $F_{j}$ . Outside $O_{j} ∖ F_{j}$ the functions $ψ_{j}$ and $1_{E_{j}}$ agree; everywhere their difference has magnitude at most one. Therefore $∥ ψ_{j} - 1_{E_{j}} ∥_{L^{p} (μ)}^{p} \leq μ (O_{j} ∖ F_{j}) < η$ .

3.1step 1.1step 2.1F5choose∎

The triangle inequality [F5] gives ${∥ \sum_{j = 1}^{N} c_{j} ψ_{j} - s ∥}_{L^{p} (μ)} \leq \sum_{j = 1}^{N} ∣ c_{j} ∣ η^{1 / p}$ . Choose $η$ small enough that this is below $ε / 2$ , and choose $s$ from step 1.1 within $ε / 2$ of $f$ . Then the compactly supported continuous function $\sum_{j} c_{j} ψ_{j}$ is within $ε$ of $f$ in $L^{p} (μ)$ .

Depends on

Used by

Single-hidden-layer universal approximation Theorem

Dependency tree · two levels

30 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

Leshno, Lin, Pinkus, and Schocken, Multilayer Feedforward Networks With a Non-Polynomial Activation Function Can Approximate Any Function, proof of Theorem 1, Lp part (standard reference, not scraped)
Kurt Hornik, Approximation Capabilities of Multilayer Feedforward Networks, §2 (standard reference, not scraped)