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Integral invariance under measure-preserving maps
Statement
If preserves and is measurable, then , allowing infinity. If is integrable real or complex valued, is integrable and the same equality holds. Conversely, for a measurable self-map, equality for every measurable indicator implies measure preservation.
Facts & Assumptions
Nonnegative measurable functions admit increasing simple approximations Every nonnegative measurable function is the increasing limit of simple measurable functions.
Increasing nonnegative measurable functions have increasing integrals with the expected limit Monotone convergence for the integral.
The integral is complex-linear on integrable functions The Lebesgue integral is linear on .
Proof
Given: The objects and hypotheses in the statement.
For , , hence its integral is . A nonnegative simple function written over disjoint fibers has integral equal to the sum of coefficient times fiber measure, so the identity holds for every such function, also when the sum is infinite in value.
For nonnegative measurable , take as supplied by simple approximation. Then measurably. Integral monotone convergence on both sides gives .
For integrable , applying step 2.1 to proves . Apply that step to the positive and negative parts of each real component. Subtract their finite integrals and combine the real and imaginary parts by linearity to obtain the asserted equality.
Conversely the indicator identity is exactly for each measurable . Thus it is measure preservation.
Depends on
Used by
- Mass conservation and positivity of the heat flow Corollary
- A nonintegrable observable with divergent ergodic averages Counterexample
- An orientation-reversing homeomorphism need not be quasiconformal Counterexample
- Irrational rotation is ergodic but not weakly mixing Counterexample
- The fundamental Hessian is not absolutely locally integrable Counterexample
- Reiter's condition (P1) Definition
- The one-dimensional torus and its normalized Haar integral Definition
- A circle representation with an averaged orthogonal weight form Example
- A symmetric random walk on the circle Example
- Affine flux reduces the entropy semigroup to translation Example
- Compact groups have a constant Reiter net Example
- Heat evolution of affine and quadratic polynomials Example
- Matching C¹ pieces across a hyperplane have no jump derivative Example
- Newton shell theorem from harmonic mean values Example
- Opposed most powerful tests for two normal alternatives Example
- The positive-type Gaussian on the real line and its cyclic model Example
- Birkhoff's theorem requires integrability False statement
- A positive rank-one Haar average is a nonzero compact intertwiner Lemma
- A Reiter net has an invariant-mean cluster point Lemma
- A translation-invariant L1 function on the line is zero Lemma
- Averaging a Hermitian form unitarizes a finite-dimensional compact-group representation Lemma
- Compact convolution operators commute with right translations and have conjugate-kernel adjoints Lemma
- Compactly supported smooth functions are dense in Slobodeckij spaces Lemma
- Convolution on L¹(ℝⁿ) is independent of the chosen Borel representatives Lemma
- First and second Gaussian heat-kernel moments Lemma
- Haar averaging projects contractively onto the bounded intertwiners Lemma
- L² convolution on a compact group is Hilbert–Schmidt Lemma
- Matrix coefficients of finite-dimensional representations separate points of a compact group Lemma
- Monomial integrals on the sphere and orthonormality on the distinguished torus Lemma
- Monomials form complete orthogonal systems of the Bergman spaces of the disc, the ball and the polydisc Lemma
- The coordinate-direction form of the Slobodeckij seminorm Lemma
- The heat kernel semigroup identity Γₜ*Γₛ=Γₜ₊ₛ Lemma
- The L1 action of a strongly continuous unitary representation Lemma
- The maximal ergodic inequality on a probability space Lemma
- The Schrödinger Weyl system is regular and irreducible Lemma
- Triangular averages of finite-past information converge to the infinite-past average Lemma
- Compact and locally compact abelian groups are amenable Proposition
- Convolution on L¹(ℝⁿ) is bilinear, commutative, and associative Proposition
- Ergodic averages are measurable, representative independent, and Lp contractive Proposition
- A two-sided normal alternative need not have a UMP test Theorem
…and 19 more results.
Dependency tree · two levels
21 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- E–W Lemma 2.6, pp.15–16 (standard reference, not scraped)