Math 2.0

Measure and integration Measures on topological spaces Theorem

Minlos theorem

Builds on Nuclear spaceDual spaceProbability measureCharacteristic functionCylinder set measure

Statement

Let EE be a real nuclear space that is also a Fréchet space, and let E′E' be its topological dual space. Give E′E' the σ-algebra generated by the maps x↦x(f)x \mapsto x(f) for f∈Ef \in E. Let φ ⁣:E→ℂ\varphi \colon E \to \C be a function. Then there is a probability measure μ\mu on E′E' with φ(f)=∫E′eix(f) dμ(x)for all f∈E\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. φ\varphi is continuous.
  2. φ\varphi is positive-definite: ∑j,k=1ncjck‾ φ(fj−fk)≥0\sum_{j,k=1}^n c_j \overline{c_k} \, \varphi(f_j - f_k) \ge 0 for all nn, all f1,…,fn∈Ef_1, \dots, f_n \in E and all c1,…,cn∈ℂc_1, \dots, c_n \in \C.
  3. φ(0)=1\varphi(0) = 1.

When μ\mu exists, it is unique.

Remarks

The function φ\varphi is the characteristic functional of μ\mu, the infinite-dimensional analogue of a characteristic function. Necessity of the three conditions is direct. For sufficiency, Bochner's theorem on each finite-dimensional quotient gives a cylinder set measure on E′E'. Nuclearity of EE is what makes this cylinder set measure countably additive.

Example: on the Schwartz space E=𝒮(ℝn)E = \mathcal{S}(\R^n), the function φ(f)=exp⁡(−12∫ℝnf(t)2 dt)\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 𝒮′(ℝn)\mathcal{S}'(\R^n) is white noise.

Nuclearity cannot be dropped. On an infinite-dimensional real Hilbert space HH, the function exp⁡(−12∥f∥2)\exp(-\frac12 \norm{f}^2) is continuous, positive-definite and equal to 11 at 00. Yet there is no probability measure on H′≅HH' \cong H with this characteristic functional.

Going deeper

For a student

The theorem is the standard way to construct Gaussian and other random fields as random elements of 𝒮′(ℝn)\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.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph
Characteristic function, concept, no evidence levelCharacteristic functi…conceptCylinder set measure, concept, no evidence levelCylinder set measureconceptDual space, concept, no evidence levelDual spaceconceptNuclear space, concept, no evidence levelNuclear spaceconceptProbability measure, concept, no evidence levelProbability measureconceptMinlos theorem, statement, no evidence levelMinlos theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Characteristic function concept
  2. Cylinder set measure concept
  3. Dual space concept
  4. Nuclear space concept
  5. Probability measure concept
  6. Minlos theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Minlos%27s_theorem
About this page
TagS2J9CBPJ0V
URIm2://S2J9CBPJ0V
Kindstatement, theorem
WritingW0 stub
MSC28C20, 46A11, 60B11
StatusThis copy of the site has no build data. Run m2 status S2J9CBPJ0V.
AboutNuclear space, Cylinder set measure
UsesNuclear space, Dual space, Probability measure, Characteristic function, Cylinder set measure
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Changed2 November 2026

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