operator induced by a measure preserving map


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1 Induced Operators

Let (X1,𝔅1,μ1) and (X2,𝔅2,μ2) be measure spacesMathworldPlanetmath and denote by L0(X1) and L0(X2) the corresponding spaces of measurable functionsMathworldPlanetmath (with values in ).

Definition - If T:X1X2 is a measure-preserving transformation we can define an operator UT:L0(X2)L0(X1) by


The operator UT is called the by T.

Many ideas in ergodic theory can be explored by studying this operator.

2 Basic Properties

The following are clear:

  • UT is linear.

  • UT maps real valued functions to real valued functions.

  • If f0 then UTf0

  • UTk=k for every constant function k.

  • UT(fg)=UT(f)UT(g).

  • UT maps characteristic functionsMathworldPlanetmathPlanetmathPlanetmathPlanetmath to characteristic functions. Moreover, UTχB=χT-1B, for every measurable setMathworldPlanetmath B𝔅2.

  • If T1:X1X2 and T2:X2X3 are measure preserving maps, then UT2T1=UT1UT2.

3 Preserving Integrals

Theorem 1 - If fL0(X2) then X1UTf𝑑μ1=X2f𝑑μ2, where if one side does not exist or is infiniteMathworldPlanetmathPlanetmath, then the other side has the same property.

4 Induced Isometries

It can further be seen that a measure-preserving transformation induces an isometry between Lp-spaces (http://planetmath.org/LpSpace), for p1.

Theorem 2 - Let p1. We have that UT(Lp(X2))Lp(X1) and, moreover,

UT(f)p=fp,for allfLp(X2)

Thus, when restricted to Lp-spaces, UT is called the isometry induced by T.

Title operator induced by a measure preserving map
Canonical name OperatorInducedByAMeasurePreservingMap
Date of creation 2013-03-22 17:59:19
Last modified on 2013-03-22 17:59:19
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 7
Author asteroid (17536)
Entry type Definition
Classification msc 47A35
Classification msc 28D05
Classification msc 37A05
Defines isometry induced by a measure preserving map