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On a semifinite measure space, a representing $L^{q}$ function is unique

Statement

Let $(X, A, μ)$ be a semifinite measure space, let $1 \leq p < \infty$ , and let $q$ be conjugate to $p$ . If $g, h \in L^{q} (μ)$ satisfy $\int f g d μ = \int f h d μ ([f] \in L^{p} (μ)),$ then $g = h$ in $L^{q} (μ)$ . Equivalently, on a semifinite measure space a bounded functional on $L^{p} (μ)$ has at most one representing $L^{q} (μ)$ class.

Facts & Assumptions

Given: A semifinite measure space $(X, A, μ)$ , an exponent $1 \leq p < \infty$ , its conjugate exponent $q$ , and $g, h \in L^{q} (μ)$ such that $\int f g d μ = \int f h d μ$ for every $[f] \in L^{p} (μ)$ .

[L1]

On a semifinite measure space, $∥ Λ_{u} ∥ = ∥ u ∥_{q} (u \in L^{q} (μ))$ (The functional $Λ_{g}$ has norm $∥ g ∥_{q}$ ; for $q = \infty$ assume $μ$ is semifinite).

[L2]

For finite exponents, the $L^{q}$ norm is a genuine norm on the quotient space The $L^{p}$ norm descends to the quotient and makes $L^{p}$ a normed space for $1 \leq p \leq \infty$ .

[L3]

If $∥ u ∥_{\infty} = 0$ , then $∣ u ∣ \leq 0$ almost everywhere, so $u = 0$ almost everywhere (The essential supremum is attained as the least essential bound).

Proof

Proof technique: Subtract the two representing functions, observe that the induced pairing is the zero functional, and use the norm formula for $Λ_{g}$ to force the difference to be the zero $L^{q}$ class.

1.1

For every $[f] \in L^{p} (μ)$ , the assumption gives [given, algebra] $Λ_{g - h} ([f]) = \int f (g - h) d μ = 0.$ So $Λ_{g - h}$ is the zero functional on $L^{p} (μ)$ .

givenalgebra

2.1

Applying [L1] to $u = g - h$ yields [L1, step 1.1] $∥ g - h ∥_{q} = ∥ Λ_{g - h} ∥ = 0.$

3.1

If $q < \infty$ , [L2] says that zero $L^{q}$ norm means the zero class, so [L2, L3, step 2.1] $g = h$ in $L^{q} (μ)$ . If $q = \infty$ , then step 2.1 and [L3] give $g = h$ almost everywhere. In either case the representing class is unique. ∎

Depends on

Used by

Dependency tree · two levels

15 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, Proposition 7.13 and Theorem 7.14 (standard reference, not scraped)

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