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For $1 \leq p < \infty$ , every $L^{p} (μ)$ class has a sigma-finite essential support

Statement

Let $(X, A, μ)$ be a measure space and let $1 \leq p < \infty$ . For every element $[f] \in L^{p} (μ)$ there is a measurable sigma-finite set $S \subseteq X$ such that $f = 0$ almost everywhere on $X ∖ S$ .

Facts & Assumptions

Given: A measure space $(X, A, μ)$ , an exponent $1 \leq p < \infty$ , and a class $[f] \in L^{p} (μ)$ .

[L1]

Elements of $L^{p} (μ)$ are almost-everywhere classes of measurable representatives (The space $L^{p} (μ)$ as the quotient by null functions).

[L2]

Chebyshev-Markov gives $μ ({∣ u ∣^{p} \geq t}) \leq t^{- 1} \int ∣ u ∣^{p} d μ (t > 0)$ (Chebyshev-Markov inequality for the integral).

[L3]

Sigma-finiteness means a countable union of finite-measure measurable sets (Finite, sigma-finite, and semifinite measures).

Proof

Proof technique: Take a representative $u$ and use the level sets ${∣ u ∣ \geq 1 / n}$ . Chebyshev-Markov makes each level set finite-measure, and their union contains every point where $u \neq 0$ .

1.1

Choose a measurable representative $u$ of $[f]$ and, for each $n \geq 1$ , [L1, L2, given, choose, construct] set $E_{n} : = {∣ u ∣ \geq 1 / n} = {∣ u ∣^{p} \geq n^{- p}} .$ Since $u \in L^{p} (μ)$ , [L2] gives $μ (E_{n}) \leq n^{p} \int ∣ u ∣^{p} d μ < \infty .$ So every $E_{n}$ has finite measure.

2.1

Put [L3, step 1.1] $S : = ⋃_{n = 1}^{\infty} E_{n} .$ By step 1.1 and [L3], the set $S$ is sigma-finite.

3.1

If $x \notin S$ , then $∣ u (x) ∣ < 1 / n$ for every $n \geq 1$ , hence $u (x) = 0$ . [step 2.1, algebra] Therefore $u = 0$ on $X ∖ S$ , so the class $[f]$ vanishes almost everywhere outside the sigma-finite set $S$ . ∎

Depends on

Used by

For 1 < p < ∞, the same representation theorem holds on arbitrary measure spaces Theorem

Dependency tree · two levels

14 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

Gerald B. Folland, Real Analysis, 2nd ed., Theorem 6.15 (standard reference, not scraped)
John K. Hunter, Measure Theory, Theorem 7.14 (standard reference, not scraped)

For 1≤p<∞1≤p<∞, every Lp(μ)Lp(μ) class has a sigma-finite essential support