operator induced by a measure preserving map
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1 Induced Operators
Let $({X}_{1},{\U0001d505}_{1},{\mu}_{1})$ and $({X}_{2},{\U0001d505}_{2},{\mu}_{2})$ be measure spaces^{} and denote by ${L}^{0}({X}_{1})$ and ${L}^{0}({X}_{2})$ the corresponding spaces of measurable functions^{} (with values in $\u2102$).
Definition  If $T:{X}_{1}\u27f6{X}_{2}$ is a measurepreserving transformation we can define an operator ${U}_{T}:{L}^{0}({X}_{2})\u27f6{L}^{0}({X}_{1})$ by
$$({U}_{T}f)(x):=f(Tx),f\in {L}^{0}({X}_{2}),x\in {X}_{1}$$ 
The operator ${U}_{T}$ is called the by $T$.
Many ideas in ergodic theory can be explored by studying this operator.
2 Basic Properties
The following are clear:

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${U}_{T}$ is linear.

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${U}_{T}$ maps real valued functions to real valued functions.

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If $f\ge 0$ then ${U}_{T}f\ge 0$

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${U}_{T}k=k$ for every constant function $k$.

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${U}_{T}(fg)={U}_{T}(f){U}_{T}(g)$.

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${U}_{T}$ maps characteristic functions^{} to characteristic functions. Moreover, ${U}_{T}{\chi}_{B}={\chi}_{{T}^{1}B}$, for every measurable set^{} $B\in {\U0001d505}_{2}$.

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If ${T}_{1}:{X}_{1}\u27f6{X}_{2}$ and ${T}_{2}:{X}_{2}\u27f6{X}_{3}$ are measure preserving maps, then ${U}_{{T}_{2}\circ {T}_{1}}={U}_{{T}_{1}}{U}_{{T}_{2}}$.
3 Preserving Integrals
Theorem 1  If $f\in {L}^{0}({X}_{2})$ then ${\int}_{{X}_{1}}{U}_{T}f\mathit{d}{\mu}_{1}={\int}_{{X}_{2}}f\mathit{d}{\mu}_{2}$, where if one side does not exist or is infinite^{}, then the other side has the same property.
4 Induced Isometries
It can further be seen that a measurepreserving transformation induces an isometry between ${L}^{p}$spaces (http://planetmath.org/LpSpace), for $p\ge 1$.
Theorem 2  Let $p\ge 1$. We have that ${U}_{T}({L}^{p}({X}_{2}))\subseteq {L}^{p}({X}_{1})$ and, moreover,
$${\parallel {U}_{T}(f)\parallel}_{p}={\parallel f\parallel}_{p},\text{for all}f\in {L}^{p}({X}_{2})$$ 
$$
Thus, when restricted to ${L}^{p}$spaces, ${U}_{T}$ is called the isometry induced by $T$.
Title  operator induced by a measure preserving map 

Canonical name  OperatorInducedByAMeasurePreservingMap 
Date of creation  20130322 17:59:19 
Last modified on  20130322 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 