operator induced by a measure preserving map

theoreminducedpropertypropertiesside

1 Induced Operators

Let $(X_1,{𝔅}_1,{\mu }_1)$ and $(X_2,{𝔅}_2,{\mu }_2)$ be measure spaces and denote by $L^0(X_1)$ and $L^0(X_2)$ the corresponding spaces of measurable functions (with values in $ℂ$).

Definition - If $T:X_1⟶X_2$ is a measure-preserving transformation we can define an operator $U_T:L^0(X_2)⟶L^0(X_1)$ 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:

• $U_T$ is linear.

• $U_T$ maps real valued functions to real valued functions.

• If $f\ge 0$ then $U_{T}f\ge 0$

• $U_{T}k=k$ 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 $B\in {𝔅}_2$.

• If $T_1:X_1⟶X_2$ and $T_2:X_2⟶X_3$ 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 ${\int }_{X_1}{U}_{T}f𝑑{\mu }_1={\int }_{X_2}f𝑑{\mu }_2$, 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 $p\ge 1$.

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 $T$.