Math 2.0

Measure and integration Definition

Gaussian measure

Builds on Probability measureLebesgue measureDirac measureBanach spaceSeparable spaceContinuous linear mapPushforward measureNormal distribution

Definition

Let γ\gamma be a Borel probability measure on ℝ\R. It is Gaussian in two cases. The first case is γ=δm\gamma = \delta_m, the Dirac measure at some m∈ℝm \in \R. The second case is that there are m∈ℝm \in \R and σ>0\sigma > 0 with γ(A)=12πσ2∫Ae−(x−m)2/(2σ2) dx\gamma(A) = \frac{1}{\sqrt{2 \pi \sigma^2}} \int_A e^{-(x - m)^2 / (2 \sigma^2)} \, dx for every Borel set A⊆ℝA \subseteq \R, where dxdx is Lebesgue measure. Let EE be a real separable Banach space. A Borel probability measure γ\gamma on EE is Gaussian if for every continuous linear form ℓ ⁣:E→ℝ\ell \colon E \to \R the pushforward ℓ∗γ\ell_* \gamma is a Gaussian measure on ℝ\R. It is centered if every ℓ∗γ\ell_* \gamma has mean 00.

Remarks

In the second case on ℝ\R, γ\gamma is the normal distribution N(m,σ2)N(m, \sigma^2); the Dirac case is the degenerate normal distribution N(m,0)N(m, 0). The standard Gaussian measure on ℝn\R^n has density (2π)−n/2e−∣x∣2/2(2\pi)^{-n/2} e^{-\abs{x}^2/2}. Some authors call γ\gamma Gaussian only when every ℓ∗γ\ell_* \gamma with ℓ≠0\ell \ne 0 is non-degenerate.

Example: on E=ℝ2E = \R^2, the standard Gaussian measure is Gaussian. For ℓ(x)=ax1+bx2\ell(x) = a x_1 + b x_2, the law ℓ∗γ\ell_* \gamma is N(0,a2+b2)N(0, a^2 + b^2), since a linear combination of independent standard normal variables is normal.

Non-example: the uniform probability measure on [−1,1][-1, 1] is not Gaussian. Its pushforward under the identity is not a Dirac measure and has no Gaussian density, since it vanishes outside [−1,1][-1, 1].

Wiener measure on the Banach space of continuous functions ω\omega on [0,1][0, 1] with ω(0)=0\omega(0) = 0 is a centered Gaussian measure.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

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Banach space, concept, no evidence levelBanach spaceconceptBorel sigma-algebra, concept, no evidence levelBorel sigma-algebraconceptContinuous linear map, concept, no evidence levelContinuous linear mapconceptDirac measure, concept, no evidence levelDirac measureconceptEuclidean space, object, no evidence levelEuclidean spaceobjectNormal distribution, concept, no evidence levelNormal distributionconceptOuter measure, concept, no evidence levelOuter measureconceptProbability measure, concept, no evidence levelProbability measureconceptPushforward measure, concept, no evidence levelPushforward measureconceptSeparable space, concept, no evidence levelSeparable spaceconceptLebesgue measure, object, no evidence levelLebesgue measureobjectGaussian measure, definition, no evidence levelGaussian measuredefinition
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. Banach space concept
  2. Borel sigma-algebra concept
  3. Continuous linear map concept
  4. Dirac measure concept
  5. Euclidean space object
  6. Normal distribution concept
  7. Outer measure concept
  8. Probability measure concept
  9. Pushforward measure concept
  10. Separable space concept
  11. Lebesgue measure object
  12. Gaussian measure definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Gaussian_measure
About this page
TagNXT0V2QWSC
URIm2://NXT0V2QWSC
Kinddefinition
WritingW0 stub
MSC28C20, 60G15, 60B11
DefinesGaussian measure
UsesProbability measure, Lebesgue measure, Dirac measure, Banach space, Separable space, Continuous linear map, Pushforward measure, Normal distribution
Changed2 November 2026

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