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$C_{c}^{\infty} (R^{n})$ is dense in $L^{p} (R^{n})$ for $1 \leq p < \infty$

Statement

Assume the Axiom of Countable Choice.

Let $1 \leq p < \infty$ . Then $C_{c}^{\infty} (R^{n})$ is 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]

$C_{c} (R^{n})$ is dense in $L^{p} (R^{n})$ ( $C_{c} (R^{n})$ is dense in $L^{p} (R^{n})$ for $1 \leq p < \infty$ ).

[L2]

Mollifier families are $L^{1}$ approximate identities, and convolution with a mollifier is smooth (A unit-mass smooth bump generates an $L^{1}$ approximate identity, Convolution with a mollifier is smooth, and derivatives pass under the integral sign).

[L3]

Approximate identities converge in $L^{p}$ , and for bounded continuous functions the convergence is uniform on compacta (Every $L^{1}$ approximate identity converges to the identity in $L^{p}$ for $1 \leq p < \infty$ , $L^{1}$ approximate identities converge uniformly on compacta for bounded continuous functions).

[L4]

The support of a convolution lies in the closure of the support sumset (The support of a convolution lies in the closure of the support sumset).

Proof

technique · direct

1.1L1L2givenchoose

By [L1], choose $g \in C_{c} (R^{n})$ with $∥ f - g ∥_{p} < ε / 2$ . [L1, L2, given, choose] Let $(φ_{δ})$ be a mollifier family as in [L2].

2.1L2L3L4step 1.1choose

By [L3], choose $δ > 0$ so small that [L2, L3, L4, step 1.1, choose] $∥ g * φ_{δ} - g ∥_{p} < ε / 2$ . The function $g * φ_{δ}$ is smooth by [L2]. Because $g$ has compact support and $φ_{δ}$ has compact support, [L4] gives compact support for $g * φ_{δ}$ , so $g * φ_{δ} \in C_{c}^{\infty} (R^{n})$ .

3.1step 1.1step 2.1algebra∎

Therefore [step 1.1, step 2.1, algebra] $∥ f - g * φ_{δ} ∥_{p} \leq ∥ f - g ∥_{p} + ∥ g - g * φ_{δ} ∥_{p} < ε .$ Hence $C_{c}^{\infty} (R^{n})$ is dense in $L^{p} (R^{n})$ .

Depends on

Dependency tree · 2 levels

Sources

Richard L. Wheeden and Antoni Zygmund, Measure and Integral: An Introduction to Real Analysis