Math 2.0

Measure and integration Definition

Quasi-invariant measure

Builds on MeasureGroup actionNull setPushforward measureAbsolutely continuous measureGaussian measureDirac measureCameron–Martin theorem

Definition

Let (X,Σ)(X, \Sigma) be a measurable space and let μ\mu be a measure on Σ\Sigma. Let GG be a group that acts on XX by group action, such that each map Tg ⁣:X→XT_g \colon X \to X, x↦g⋅xx \mapsto g \cdot x, is measurable. The measure μ\mu is quasi-invariant under GG if for every g∈Gg \in G and every A∈ΣA \in \Sigma, μ(A)=0  ⟺  μ(Tg−1(A))=0.\mu(A) = 0 \iff \mu(T_g^{-1}(A)) = 0.

Remarks

Equivalently, for each gg the pushforward (Tg)∗μ(T_g)_* \mu and μ\mu are each absolutely continuous with respect to the other. Every invariant measure is quasi-invariant.

Example: let G=ℝG = \R act on X=ℝX = \R by translations, and let μ\mu be the standard Gaussian measure. Its density is positive everywhere, so μ(A)=0\mu(A) = 0 exactly when AA has Lebesgue measure 00. Lebesgue null sets are preserved by translations, so μ\mu is quasi-invariant.

Non-example: under the same action, the Dirac measure δ0\delta_0 is not quasi-invariant. For g=1g = 1 and A={1}A = \set{1}, one has δ0(A)=0\delta_0(A) = 0 but δ0(T1−1(A))=δ0({0})=1\delta_0(T_1^{-1}(A)) = \delta_0(\set{0}) = 1.

By the Cameron–Martin theorem, Wiener measure is quasi-invariant under translations by the Cameron–Martin space, but not under all translations of the path space.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

Show the path as a graph
Absolutely continuous measure, concept, no evidence levelAbsolutely continuous…conceptClassical Wiener space, object, no evidence levelClassical Wiener spaceobjectDirac measure, concept, no evidence levelDirac measureconceptGaussian measure, concept, no evidence levelGaussian measureconceptGroup action, concept, no evidence levelGroup actionconceptMeasure, concept, no evidence levelMeasureconceptMutually singular measures, concept, no evidence levelMutually singular mea…conceptNull set, concept, no evidence levelNull setconceptPushforward measure, concept, no evidence levelPushforward measureconceptRadon–Nikodym derivative, concept, no evidence levelRadon–Nikodym derivat…conceptWiener process, concept, no evidence levelWiener processconceptCameron–Martin theorem, statement, no evidence levelCameron–Martin theoremstatementQuasi-invariant measure, definition, no evidence levelQuasi-invariant measu…definition
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. Absolutely continuous measure concept
  2. Classical Wiener space object
  3. Dirac measure concept
  4. Gaussian measure concept
  5. Group action concept
  6. Measure concept
  7. Mutually singular measures concept
  8. Null set concept
  9. Pushforward measure concept
  10. Radon–Nikodym derivative concept
  11. Wiener process concept
  12. Cameron–Martin theorem statement
  13. Quasi-invariant measure definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Quasi-invariant_measure
About this page
TagEGQ4GHH127
URIm2://EGQ4GHH127
Kinddefinition
WritingW0 stub
MSC28C10, 28D05, 22D40
DefinesQuasi-invariant measure
UsesMeasure, Group action, Null set, Pushforward measure, Absolutely continuous measure, Gaussian measure, Dirac measure, Cameron–Martin theorem
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.