MathDB

6. Let

T

be a one-to-one mapping of the unit square

E

of the plane into itself. Suppose that

T

and

T^{-1}

are measure-preserving (i.e. if

M \subseteq E

is a measurable set, then

TM

and

T^{-1}M

are also measurable and

\mu (M)= \mu (TM)= \mu (T^{-1}M)

, where

\mu

denotes the Lebesgue measure) and, furthermore, that if

Tx \in N

for almost all points

x

of a measurable set

N \subseteq E

, then either

n

E \setminus N

is of measure 0. Prove that, for any measurable set

A \subseteq E

, with

\mu (A)>0

, the function

n(x)

defined by
n(x)=\begin{cases} 0, \mbox{if} T^k x \notin A (k=1, 2, \dots),\\ \min (k: T^k x \in A; k=1,2, \dots ) &\mbox{otherwise} \end{cases}
is measurable and

\int_{A}n(x) d\mu(x) =1

(R. 18)

Problem Statement