Hermitian Measures and Complex Measures

Introduction

A complex measure is a countable additive set function with complex values, and the theory of the integral associates with it its total variation $|\mu|$, a positive measure, and the polar decomposition $\mu=h\,|\mu|$ with $\lvert h\rvert=1$ almost everywhere, which is the measure-theoretic counterpart of the polar form $z=\lvert z\rvert e^{i\theta}$ of a complex number. The operation of complex conjugation acts on the measures by conjugating their values, $$ \bar\mu(A)=\overline{\mu(A)}, $$ and this involution is the subject of the present article: the measures fixed by it are the Hermitian ones, exactly the real signed measures; an arbitrary complex measure splits into a Hermitian part and an anti-Hermitian part; and the polar decomposition is compatible with the involution by the conjugation of the density. The involution is the same involution as that of the previous articles of this group — complex conjugation of functions — now on the space of measures, and the measures that are both Hermitian and positive are exactly the ones that Bochner's theorem produces from the positive definite functions.

This article is the third of the * group of Foundations of Analysis. Its prerequisites are Measure Theory and Integration for the signed and complex measures, the Hahn–Jordan decomposition, the total variation, the Radon–Nikodym theorem and the convergence theorems, and Positive Definite Functions and Hermitian Kernels, the previous article of this group, for Bochner's theorem and the positive definite functions. The Fourier–Stieltjes transform of a measure and its compatibility with the involution use the Fourier transform of Fourier Analysis on Euclidean Spaces; the general measure algebra of a locally compact group and the convolution of measures are Convolution on a Group and Harmonic Analysis on Groups, in the neighbouring category Analysis on Groups of this Part, and are named as forward references only. The Riesz representation theorem that produces a measure from a positive functional is quoted from Measure Theory and Integration, as in the previous article. The positive definite distributions of the next article are the dual form of the present material; the Hermitian kernels of integral operators are the * Operator article later in this group. No geometry is invoked.

The Space of Complex Measures

Definition and the Total Variation

Definition. Let $(X,\mathcal A)$ be a measurable space. A complex measure is a function $\mu:\mathcal A\to\mathbb C$ with $\mu(\varnothing)=0$ that is countably additive; a signed measure is the real-valued analogue. The total variation of a complex measure is the nonnegative set function $$ \lvert\mu\rvert(A)=\sup\Bigl\{\sum_{j=1}^n\lvert\mu(A_j)\rvert : A=\bigsqcup_{j=1}^nA_j,\ A_j\in\mathcal A\Bigr\} , $$ the supremum over the finite measurable partitions of $A$; the variation norm is $\lVert\mu\rVert=\lvert\mu\rvert(X)$.

Theorem (the variation is a measure and the norm is complete). For every complex measure $\mu$ the set function $\lvert\mu\rvert$ is a finite positive measure; every complex measure is a finite linear combination of positive finite measures, $$ \mu=\mu_1-\mu_2+i(\mu_3-\mu_4), $$ and the space $M(X)$ of complex measures is a complex vector space in which the variation norm is complete, so $M(X)$ is a Banach space.

Proof. The Hahn–Jordan decomposition of Measure Theory and Integration writes a real signed measure as $\nu=\nu^+-\nu^-$ with $\lvert\nu\rvert=\nu^++\nu^-$, and applying it to the real and imaginary parts of $\mu$ gives the four-measure representation; the variation of the combination is at most the sum of the variations, so $M(X)$ embeds in the product of four copies of the space of finite measures, closed under the norm; completeness follows from the completeness of the space of finite measures in the variation norm, which is the countable additivity under limits. $\blacksquare$

The Involution

Definition. The involution on $M(X)$ is $$ \bar\mu(A)=\overline{\mu(A)},\qquad A\in\mathcal A . $$ It is additive, conjugate-linear over $\mathbb C$, involutive, and it is an isometry for the variation norm, $\lvert\bar\mu\rvert=\lvert\mu\rvert$ and $\lVert\bar\mu\rVert=\lVert\mu\rVert$; with this involution $M(X)$ is a Banach space carrying an involutive conjugate-linear isometry.

Proof. Additivity and conjugation are immediate; $\lvert\bar\mu\rvert(A)=\lvert\mu\rvert(A)$ because a partition has the same variation for $\bar\mu$ and $\mu$; the involution is conjugate-linear by the conjugation of the scalars. $\blacksquare$

Remark (the two involutions). The involution $\bar\mu$ is the value-conjugation, the one induced by the complex conjugation of functions through the pairing $\langle\mu,f\rangle=\int f\,d\mu$ of the last section. It is not the adjoint involution $\mu^{*}(A)=\overline{\mu(A^{-1})}$ of the group measure algebra, which appears when $X$ is a group and $M(X)$ is given the convolution product; the adjoint involution is anti-multiplicative and is treated in Convolution on a Group, in the neighbouring category Analysis on Groups of this Part. Value-conjugation, by contrast, commutes with convolution, $\overline{\mu*\nu}=\bar\mu*\bar\nu$. The distinction is recorded so that the mark $\bar\mu$ is never read as the algebra involution.

Definition. A complex measure is Hermitian if $\bar\mu=\mu$, and anti-Hermitian if $\bar\mu=-\mu$; the Hermitian part and the anti-Hermitian part of $\mu$ are $$ \mu_{\mathrm h}=\frac12(\mu+\bar\mu),\qquad \mu_{\mathrm a}=\frac12(\mu-\bar\mu),\qquad \mu=\mu_{\mathrm h}+\mu_{\mathrm a} . $$ Equivalently $\mu_{\mathrm h}=\operatorname{Re}\mu$ and $\mu_{\mathrm a}=i\operatorname{Im}\mu$, where $\operatorname{Re}\mu$ and $\operatorname{Im}\mu$ are the real signed measures defined by $(\operatorname{Re}\mu)(A)=\operatorname{Re}(\mu(A))$ and $(\operatorname{Im}\mu)(A)=\operatorname{Im}(\mu(A))$.

Theorem (the Hermitian measures are the real signed measures). A complex measure $\mu$ is Hermitian if and only if $\mu(A)\in\mathbb R$ for every $A$, if and only if $\mu$ is a real signed measure; the Hermitian measures form a real vector space, and every complex measure decomposes uniquely as a Hermitian plus an anti-Hermitian measure.

Proof. $\bar\mu=\mu$ is the statement $\overline{\mu(A)}=\mu(A)$ for all $A$, that is $\mu(A)\in\mathbb R$; the converse is immediate. The decomposition is the one displayed, and its uniqueness is the uniqueness of the real and imaginary parts. $\blacksquare$

Polar Decomposition

The Polar Form of a Measure

Definition. A polar decomposition of a complex measure $\mu$ is a representation $$ \mu=h\,\lvert\mu\rvert,\qquad d\mu=h\,d\lvert\mu\rvert , $$ with $h$ a measurable function of modulus one $\lvert\mu\rvert$-almost everywhere.

Theorem (existence and uniqueness of the polar decomposition). Every complex measure $\mu$ has a polar decomposition, and it is unique in the sense that its density $h$ is unique $\lvert\mu\rvert$-almost everywhere.

Proof. The total variation is a positive measure and $\mu\ll\lvert\mu\rvert$: if $\lvert\mu\rvert(A)=0$ then every measurable subset of $A$ has $\mu$-measure zero, so $\mu(A)=0$. The Radon–Nikodym theorem of Measure Theory and Integration therefore gives a density $h=d\mu/d\lvert\mu\rvert\in L^1(\lvert\mu\rvert)$ with $d\mu=h\,d\lvert\mu\rvert$; the identity $\lvert\mu\rvert=\lvert h\rvert\,\lvert\mu\rvert$, proved by applying the Radon–Nikodym derivative to the definition of the variation, forces $\lvert h\rvert=1$ $\lvert\mu\rvert$-almost everywhere. Uniqueness is the uniqueness of the Radon–Nikodym derivative. $\blacksquare$

Compatibility with the Involution

Theorem. For a complex measure $\mu=h\,\lvert\mu\rvert$, $$ \bar\mu=\bar h\,\lvert\mu\rvert,\qquad \lvert\bar\mu\rvert=\lvert\mu\rvert , $$ and if $\nu$ is a measure with $\mu\ll\nu$, then $\bar\mu\ll\nu$ and $$ \frac{d\bar\mu}{d\nu}=\overline{\frac{d\mu}{d\nu}} . $$ The polar decomposition of $\bar\mu$ is obtained from that of $\mu$ by conjugating the density.

Proof. $\bar\mu(A)=\overline{\int_Ah\,d\lvert\mu\rvert}=\int_A\bar h\,d\lvert\mu\rvert$ by the conjugation of the integral, and $\lvert\bar h\rvert=1$; the Radon–Nikodym statement follows from the same computation with $\nu$ in place of $\lvert\mu\rvert$. $\blacksquare$

Corollary (Hermitian and anti-Hermitian densities). A measure $\mu=h\,\lvert\mu\rvert$ is Hermitian if and only if its density is real $\lvert\mu\rvert$-almost everywhere, and anti-Hermitian if and only if its density is purely imaginary; the Hermitian and anti-Hermitian parts have densities $\operatorname{Re}h$ and $i\operatorname{Im}h$.

Proof. The involution conjugates $h$, so the fixed points and the skew points of the involution are the real and imaginary densities. $\blacksquare$

Hermitian Measures and the Fourier–Stieltjes Transform

The Transform and Its Involution Property

Definition. For a finite complex measure $\mu$ on $\mathbb R^n$ the Fourier–Stieltjes transform is $$ \hat\mu(\xi)=\int_{\mathbb R^n}e^{-2\pi ix\cdot\xi}\,d\mu(x), $$ with the normalisation of Fourier Analysis on Euclidean Spaces; it is a bounded uniformly continuous function of $\xi$, with $\lVert\hat\mu\rVert_\infty\leq\lVert\mu\rVert$, and it determines $\mu$ uniquely.

Theorem. The involution commutes with the transform through the reflection of the argument, $$ \widehat{\bar\mu}(\xi)=\overline{\hat\mu(-\xi)} . $$ Consequently $\mu$ is Hermitian if and only if its transform is a Hermitian function, $\hat\mu(-\xi)=\overline{\hat\mu(\xi)}$, and $\mu$ is anti-Hermitian if and only if $\hat\mu$ is odd, with $\hat\mu(-\xi)=-\overline{\hat\mu(\xi)}$.

Proof. $\widehat{\bar\mu}(\xi)=\int e^{-2\pi ix\cdot\xi}d\bar\mu(x) =\overline{\int e^{2\pi ix\cdot\xi}d\mu(x)}=\overline{\hat\mu(-\xi)}$ by the conjugation of the integral and the sign change $x\mapsto-x$. The Hermitian condition $\hat\mu=\widehat{\bar\mu}$ is then the stated symmetry. $\blacksquare$

Positive Definite Functions and the Cone of Positive Measures

Theorem (Bochner, recast). A continuous function $\varphi$ on $\mathbb R^n$ is positive definite if and only if $\varphi(x)=\hat\mu(-x)$ for a unique finite positive measure $\mu$, equivalently $\varphi(x)=\int e^{2\pi ix\cdot\xi}d\mu(\xi)$; the Hermitian measures are exactly the measures whose transforms satisfy the Hermitian symmetry, and the positive measures among them are exactly the transforms of the positive definite functions.

Proof. This is Bochner's theorem of Positive Definite Functions and Hermitian Kernels, the previous article of this group, with the transform of the present Article; the statement that a positive measure gives a positive definite function is the computation $$ \sum_{i,j}\overline{c_i}c_j\hat\mu(x_j-x_i)=\int\Bigl\lvert\sum_ic_ie^{-2\pi ix_i\cdot\xi}\Bigr\rvert^2d\mu(\xi)\geq0 , $$ and the converse is the theorem quoted. $\blacksquare$

Thus the positive definite functions are the transforms of the positive cone in the Hermitian measures, and the Hermitian measures are the real subspace on which the involution acts trivially; the whole of Bochner's theorem is the statement that the transform identifies the positive cone of $M(\mathbb R^n)$ with the cone of positive definite functions.

The Pairing of a Measure with a Function

Theorem. Write $\langle\mu,f\rangle=\int f\,d\mu$ for $f\in L^1(\lvert\mu\rvert)$. The involution is the adjoint of the complex conjugation of the function, $$ \langle\bar\mu,f\rangle=\overline{\langle\mu,\bar f\rangle},\qquad \langle\mu,\bar f\rangle=\overline{\langle\bar\mu,f\rangle} . $$

Proof. $\langle\bar\mu,f\rangle=\int f\,d\bar\mu=\overline{\int\bar f\,d\mu}$ by the conjugation of the integral, which is the first identity; the second is its conjugate. $\blacksquare$

Order and the Positive Cone

Definition. For Hermitian (real, signed) measures define $\mu\leq\nu$ if $\nu-\mu$ is a positive measure, that is, if $(\nu-\mu)(A)\geq0$ for every $A$. A Hermitian measure is positive if $\mu\geq0$, and the positive cone is the set of positive measures.

Theorem. The positive cone is a closed convex cone in the real space of Hermitian measures, and it is generated by the measures of the form $\mathbf 1_A\,\lvert\mu\rvert$; a complex measure is positive if and only if it is Hermitian and its density in the polar decomposition is $+1$. The Hermitian measures are the span of the positive cone, and $\lvert\mu\rvert$ is the least positive measure dominating $\mu$ in the absolute-value sense.

Proof. The cone properties are immediate from positivity; the generation and the density statement follow from the polar decomposition, since $\lvert\mu\rvert=h^{-1}\mu$ with $h$ of modulus one and, when $\mu$ is positive, the density of its polar decomposition is the constant $1$. The domination property is the definition of the total variation. $\blacksquare$

Remark (the involution and the order). The involution is order preserving on the Hermitian measures: if $\mu\leq\nu$ with both Hermitian, then, $\mu$ and $\nu$ being real, $\bar\mu=\mu\leq\nu=\bar\nu$. The positive definite functions are exactly the positive cone, under the transform, and this is the content of Bochner's theorem in its sharpest form.

Summary

A complex measure is a countably additive $\mathbb C$-valued set function, and the space $M(X)$ of them is a Banach space with total variation norm carrying the involutive conjugate-linear isometry $\bar\mu(A)=\overline{\mu(A)}$; a measure is Hermitian, $\bar\mu=\mu$, exactly when it is a real signed measure, and every complex measure splits uniquely as $\mu=\mu_{\mathrm h}+\mu_{\mathrm a}$ with $\mu_{\mathrm h}=\operatorname{Re}\mu$ and $\mu_{\mathrm a}=i\operatorname{Im}\mu$. Every complex measure has a polar decomposition $\mu=h\lvert\mu\rvert$ with $\lvert h\rvert=1$ almost everywhere, unique in $h$, which is the Radon–Nikodym derivative of $\mu$ with respect to its variation, and the involution conjugates $h$ and commutes with the Radon–Nikodym derivative. The Fourier–Stieltjes transform satisfies $\widehat{\bar\mu}(\xi)=\overline{\hat\mu(-\xi)}$, so the Hermitian measures are exactly the measures whose transforms are Hermitian functions; Bochner's theorem identifies the positive measures with the positive definite functions, so that the positive definite functions are the transform of the positive cone, and the involution is the adjoint of complex conjugation of functions. The Hermitian measures form a real ordered space whose positive cone is closed, convex and spanned by the positive measures.

Summary of Notation

Symbol	Meaning
$M(X)$	Complex measures with the variation norm, a Banach space with value-conjugation
$\lvert\mu\rvert$, $\lVert\mu\rVert$	Total variation and variation norm
$\bar\mu(A)=\overline{\mu(A)}$	The involution on measures
$\mu_{\mathrm h}=\frac12(\mu+\bar\mu)$	Hermitian part, a real signed measure
$\mu_{\mathrm a}=\frac12(\mu-\bar\mu)$	Anti-Hermitian part, $i$ times a signed measure
$\mu=h\lvert\mu\rvert$	Polar decomposition, $\lvert h\rvert=1$ a.e.
$\hat\mu(\xi)=\int e^{-2\pi ix\cdot\xi}d\mu(x)$	Fourier–Stieltjes transform
$\widehat{\bar\mu}(\xi)=\overline{\hat\mu(-\xi)}$	Involution property of the transform
$\langle\mu,f\rangle=\int f\,d\mu$	Pairing of a measure with a function