---
tag: X9V1E3V145
kind: statement
role: theorem
name: F. and M. Riesz theorem
msc: [30H10, 42A16]
uses: [PNNNFJVF6W, TCYSWCTYYM]
about: [PNNNFJVF6W]
sources:
  - url: https://en.wikipedia.org/wiki/F._and_M._Riesz_theorem
same_as:
  wikidata: Q3527067
  wikipedia: F. and M. Riesz theorem
---

## Statement

Let $\mu$ be a [[complex measure|NY6TYHRGMS]] on the Borel sets of $[0, 2\pi)$, viewed as the unit circle.
For $n \in \Z$ let $\hat\mu(n) = \int e^{-int} \, d\mu(t)$.
Assume that $\hat\mu(n) = 0$ for every integer $n < 0$.
Then $\mu$ is [[absolutely continuous|P0P21KDV6E]] with respect to [[Lebesgue measure|2Z2QVWZ262]].
Moreover $d\mu = h \, dt$ for an integrable function $h$ whose [[Poisson integral|TCYSWCTYYM]] is holomorphic on the unit disc and belongs to the [[Hardy space|PNNNFJVF6W]] $H^1$.

## Remarks

Proof idea: the Poisson integral of $\mu$ is $F(z) = \frac{1}{2\pi} \sum_{n \ge 0} \hat\mu(n) z^n$, a holomorphic function.
Its integrals $\int \abs{F(r e^{it})} \, dt$ are bounded by the total variation of $\mu$, so $F \in H^1$.
A function in $H^1$ has boundary values $F^*$ with $F(r e^{it}) \to F^*(e^{it})$ in $L^1$, a fact that uses the [[inner-outer factorization|X6ZQPAQF76]].
Hence the measures $F(r e^{it}) \, dt$ converge both to $\mu$ and to $F^* \, dt$, and $\mu = F^* \, dt$.

Example: if $\mu$ is a point mass at $t = 0$, then $\hat\mu(n) = 1$ for all $n$, so the hypothesis fails, as it must since a point mass is not absolutely continuous.

A consequence: if $\Omega$ is a domain bounded by a rectifiable Jordan curve, then harmonic measure on $\partial\Omega$ and arc length have the same null sets.

## Explanations

### L1 human

A measure on the circle has Fourier coefficients, one for each integer.
If all the coefficients with negative index vanish, the measure is "analytic".
The theorem says that an analytic measure cannot be concentrated on a small set: it must have a density.
So analytic measures are much more regular than general measures.

### L2 human

The theorem identifies the analytic measures on the circle with the boundary values of $H^1$ functions.
It is a key step in the theory of [[Hardy spaces|PNNNFJVF6W]] and in the study of boundary behaviour of conformal maps.
Its consequence for rectifiable Jordan domains links the [[harmonic measure|CFD7BPM51W]] with arc length.