Weights, their associated measures, and the spaces L^p(w)

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), the principle used by the completeness statement below.

Weights. A weight on ${\mathbb{R}}^n$ is a Lebesgue measurable function $w:{\mathbb{R}}^n\to [0,\mathrm{\infty }]$ such that ${\int }_{B(x,r)}wd\lambda <\mathrm{\infty }$ for every Euclidean ball with $r>0$, and $0<w(x)<\mathrm{\infty }$ for Lebesgue-almost every $x$ (Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$, Measure-null sets and almost-everywhere statements relative to a measure). Thus a weight may vanish or be infinite only on a Lebesgue null set.

Set $\tilde{w}(x)=w(x)$ where $0<w(x)<\mathrm{\infty }$ and $\tilde{w}(x)=1$ otherwise. This positive finite-valued representative belongs to $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\mathbb{R}}^n)$ in the precise sense of A locally integrable function on ${\mathbb{R}}^n$. In all finite-valued function interfaces and reciprocal powers below, use this representative and denote it again by $w$. Its associated measure, cube integrals and almost-everywhere assertions agree with those of the original extended-valued weight.

The $w$-measure. For a Lebesgue measurable set $E$ the $w$-measure of $E$ is $w(E):={\int }_{E}wd\lambda \mathrm{.}$ The map $E\mapsto w(E)$ is countably additive by the indefinite-integral theorem (The indefinite integral of a nonnegative measurable function is a measure), so it is a measure on the Lebesgue $\sigma$-algebra; its restriction to the Borel sets is a Borel measure. It is finite on bounded sets: a bounded $E$ lies in some ball $B$, and monotonicity of the integral together with $w\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$ gives $w(E)\le {\int }_{B}wd\lambda <\mathrm{\infty }$. It is therefore a locally finite (equivalently, Radon) Borel measure: ${\mathbb{R}}^n$ is locally compact and $\sigma$-compact (${\mathbb{R}}^n$ is locally compact and $\sigma$-compact), so every open subset is $\sigma$-compact (for a proper open $U$, use, for integers $m\ge 1$, $K_m=\{x:\mathrm{\mid }x\mathrm{\mid }\le m,\mathrm{dist}(x,U^c)\ge 1\mathrm{/}m\}$: distance to the closed complement is continuous by the triangle inequality, so $K_m$ is closed and bounded, hence compact by Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, and $U={\bigcup }_m{K}_m$; for $U={\mathbb{R}}^n$ use closed balls), and Sigma-compact open sets make locally finite Borel measures regular makes the measure regular. Since ${\mathbb{R}}^n$ is $\sigma$-compact, $w$ is $\sigma$-finite as well.

Because $0<w<\mathrm{\infty }$ almost everywhere, a Lebesgue measurable set is $w$-null exactly when it is Lebesgue null: the integral of $w$ over a Lebesgue null set vanishes, and conversely ${\int }_{N}wd\lambda =0$ forces $w=0$ a.e. on $N$, hence $\lambda (N)=0$ since $w>0$ a.e.

The spaces $L^p(w)$. Fix $1\le p<\mathrm{\infty }$. For a measurable $f$ the weighted $L^p$ functional is $\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p(w)}:=({\int }_{{\mathbb{R}}^n}\mathrm{\mid }f{\mathrm{\mid }}^{p}wd\lambda {)}^{1\mathrm{/}p}\in [0,\mathrm{\infty }],$ the value $+\mathrm{\infty }$ being assigned when the integral diverges. Since $\int \mathrm{\mid }f{\mathrm{\mid }}^{p}wd\lambda =\int \mathrm{\mid }f{\mathrm{\mid }}^{p}d(w\lambda )$ and $w\lambda$ is a measure, this is the $L^p$ functional of the measure space $({\mathbb{R}}^n,w\lambda )$ in the sense of The function space ${\mathcal{L}}^p(\mu )$ for $0<p<\mathrm{\infty }$, and $L^p(w)$ is the corresponding quotient of the class of measurable $f$ with $\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p(w)}<\mathrm{\infty }$ by the functions that vanish $w$-almost everywhere (The space $L^p(\mu )$ as the quotient by null functions); complex-valued $f$ are admitted under the componentwise conventions of Complex Lp classes and Euclidean test-function conventions. Equivalently, and this is how the norm is used below, $\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p(w)}=\parallel \mathrm{\mid }f\mathrm{\mid }{\parallel }_{L^p(w)}$ and the quotient norm is well defined (The $L^p$ norm descends to the quotient and makes $L^p$ a normed space for $1\le p\le \mathrm{\infty }$); the real space is complete by Riesz-Fischer completeness of $L^p$ for $1\le p\le \mathrm{\infty }$ applied to $w\lambda$. For complex functions, real and imaginary component projections contract the norm and recombination has norm at most the sum of the component norms (Complex Holder, Minkowski, and the quotient norm). A complex Cauchy sequence therefore has two real Cauchy components with limits, whose recombination is its complex norm limit; hence the complex space is Banach as well. Because the $w$-null sets are exactly the Lebesgue null sets, membership of $L^p(w)$ and equality in $L^p(w)$ are determined by the same negligible sets as in unweighted measure theory.