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Miklós Schweitzer
1959 Miklós Schweitzer
6
6
Part of
1959 Miklós Schweitzer
Problems
(1)
Miklós Schweitzer 1959- Problem 6
Source:
11/8/2015
6. Let
T
T
T
be a one-to-one mapping of the unit square
E
E
E
of the plane into itself. Suppose that
T
T
T
and
T
−
1
T^{-1}
T
−
1
are measure-preserving (i.e. if
M
⊆
E
M \subseteq E
M
⊆
E
is a measurable set, then
T
M
TM
TM
and
T
−
1
M
T^{-1}M
T
−
1
M
are also measurable and
μ
(
M
)
=
μ
(
T
M
)
=
μ
(
T
−
1
M
)
\mu (M)= \mu (TM)= \mu (T^{-1}M)
μ
(
M
)
=
μ
(
TM
)
=
μ
(
T
−
1
M
)
, where
μ
\mu
μ
denotes the Lebesgue measure) and, furthermore, that if
T
x
∈
N
Tx \in N
T
x
∈
N
for almost all points
x
x
x
of a measurable set
N
⊆
E
N \subseteq E
N
⊆
E
, then either
n
n
n
or
E
∖
N
E \setminus N
E
∖
N
is of measure 0. Prove that, for any measurable set
A
⊆
E
A \subseteq E
A
⊆
E
, with
μ
(
A
)
>
0
\mu (A)>0
μ
(
A
)
>
0
, the function
n
(
x
)
n(x)
n
(
x
)
defined byn(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
∫
A
n
(
x
)
d
μ
(
x
)
=
1
\int_{A}n(x) d\mu(x) =1
∫
A
n
(
x
)
d
μ
(
x
)
=
1
(R. 18)
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