# operator induced by a measure preserving map

## 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.

• $U_{T}$ maps real valued functions to real valued functions.

• If $f\geq 0$ then $U_{T}f\geq 0$

• $U_{T}k=k$ for every constant function $k$.

• $U_{T}(fg)=U_{T}(f)U_{T}(g)$.

• $U_{T}$ maps characteristic functions to characteristic functions. Moreover, $U_{T}\chi_{B}=\chi_{T^{-1}B}$, for every measurable set $B\in\mathfrak{B}_{2}$.

• 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 OperatorInducedByAMeasurePreservingMap 2013-03-22 17:59:19 2013-03-22 17:59:19 asteroid (17536) asteroid (17536) 7 asteroid (17536) Definition msc 47A35 msc 28D05 msc 37A05 isometry induced by a measure preserving map