---
tag: S2J9CBPJ0V
kind: statement
role: theorem
name: Minlos theorem
msc: [28C20, 46A11, 60B11]
uses: [XC6XPMC2FQ, 07JBSZE0V4, 059TJN0NZ1, 0V5B6D6BQH, G95YDNFXQN]
about: [XC6XPMC2FQ, G95YDNFXQN]
sources:
  - url: https://en.wikipedia.org/wiki/Minlos%27s_theorem
same_as:
  wikidata: Q6867878
  wikipedia: Minlos's theorem
---

## Statement

Let $E$ be a real [[nuclear space|XC6XPMC2FQ]] that is also a Fréchet space, and let $E'$ be its topological [[dual space|07JBSZE0V4]].
Give $E'$ the σ-algebra generated by the maps $x \mapsto x(f)$ for $f \in E$.
Let $\varphi \colon E \to \C$ be a function.
Then there is a [[probability measure|059TJN0NZ1]] $\mu$ on $E'$ with
$$\varphi(f) = \int_{E'} e^{i x(f)} \, d\mu(x) \quad \text{for all } f \in E$$
if and only if these three conditions hold:

1. $\varphi$ is continuous.
2. $\varphi$ is positive-definite: $\sum_{j,k=1}^n c_j \overline{c_k} \, \varphi(f_j - f_k) \ge 0$ for all $n$, all $f_1, \dots, f_n \in E$ and all $c_1, \dots, c_n \in \C$.
3. $\varphi(0) = 1$.

When $\mu$ exists, it is unique.

## Remarks

The function $\varphi$ is the characteristic functional of $\mu$, the infinite-dimensional analogue of a [[characteristic function|0V5B6D6BQH]].
Necessity of the three conditions is direct.
For sufficiency, Bochner's theorem on each finite-dimensional quotient gives a [[cylinder set measure|G95YDNFXQN]] on $E'$.
Nuclearity of $E$ is what makes this cylinder set measure countably additive.

Example: on the Schwartz space $E = \mathcal{S}(\R^n)$, the function $\varphi(f) = \exp\left(-\frac12 \int_{\R^n} f(t)^2 \, dt\right)$ satisfies the three conditions.
The measure it gives on the space of tempered distributions $\mathcal{S}'(\R^n)$ is white noise.

Nuclearity cannot be dropped.
On an infinite-dimensional real Hilbert space $H$, the function $\exp(-\frac12 \norm{f}^2)$ is continuous, positive-definite and equal to $1$ at $0$.
Yet there is no probability measure on $H' \cong H$ with this characteristic functional.

## Explanations

### L1 human

A probability measure on $\R^n$ is determined by its characteristic function, and Bochner's theorem says which functions occur.
Minlos's theorem gives the same answer in infinite dimensions, for the dual of a nuclear space.
So to build a random distribution, it suffices to write down a suitable characteristic functional.

### L2 human

The theorem is the standard way to construct Gaussian and other random fields as random elements of $\mathcal{S}'(\R^n)$.
White noise, the Gaussian free field and Euclidean quantum fields are built this way.
It contrasts with the Hilbert space case, where by Sazonov's theorem continuity in a stronger topology is needed.