operator induced by a measure preserving map

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1 Induced Operators

Let $(X_{1},\mathfrak{B}_{1},\mu_{1})$ and $(X_{2},\mathfrak{B}_{2},\mu_{2})$ be measure spaces and denote by $L^{0}(X_{1})$ and $L^{0}(X_{2})$ the corresponding spaces of measurable functions (with values in $\mathbb{C}$ ).

Definition - If $T:X_{1}\longrightarrow X_{2}$ is a measure-preserving transformation we can define an operator $U_{T}:L^{0}(X_{2})\longrightarrow L^{0}(X_{1})$ by

(U_{T}f)(x):=f(Tx)\,,\qquad\qquad 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:

•

$U_{T}$ is linear.
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$U_{T}$ maps real valued functions to real valued functions.
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If $f\geq 0$ then $U_{T}f\geq 0$
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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 $B\in\mathfrak{B}_{2}$ .
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If $T_{1}:X_{1}\longrightarrow X_{2}$ and $T_{2}:X_{2}\longrightarrow X_{3}$ 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 $\int_{X_{1}}U_{T}f\;d\mu_{1}=\int_{X_{2}}f\;d\mu_{2}$ , 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 measure-preserving transformation induces an isometry between $L^{p}$ -spaces (http://planetmath.org/LpSpace), for $p\geq 1$ .

Theorem 2 - Let $p\geq 1$ . We have that $U_{T}(L^{p}(X_{2}))\subseteq L^{p}(X_{1})$ and, moreover,

\|U_{T}(f)\|_{p}=\|f\|_{p}\,,\qquad\qquad\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	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