MathDB
Problems
Contests
Undergraduate contests
Miklós Schweitzer
2010 Miklós Schweitzer
10
10
Part of
2010 Miklós Schweitzer
Problems
(1)
Finite number of Borel sets
Source: Miklós Schweitzer P10
9/9/2020
Consider the space
{
0
,
1
}
N
\{0,1 \} ^{N}
{
0
,
1
}
N
with the product topology (where
{
0
,
1
}
\{0,1 \}
{
0
,
1
}
is a discrete space). Let
T
:
{
0
,
1
}
N
→
{
0
,
1
}
N
T: \{0,1 \} ^ {\mathbb {N}} \rightarrow \{0,1 \} ^ {\mathbb {N}}
T
:
{
0
,
1
}
N
→
{
0
,
1
}
N
be the left-shift, ie
(
T
x
)
(
n
)
=
x
(
n
+
1
)
(Tx) (n) = x (n+1)
(
T
x
)
(
n
)
=
x
(
n
+
1
)
for every
n
∈
N
n \in \mathbb {N}
n
∈
N
. Can a finite number of Borel sets be given:
B
1
,
…
,
B
m
⊂
{
0
,
1
}
N
B_ {1}, \ldots, B_ {m} \subset \{0,1 \} ^ {N}
B
1
,
…
,
B
m
⊂
{
0
,
1
}
N
such that
{
T
i
(
B
j
)
∣
i
∈
N
,
1
≤
j
≤
m
}
\left \{T ^ {i} \left (B_ {j} \right) \mid i \in \mathbb {N}, 1 \leq j \leq m \right \}
{
T
i
(
B
j
)
∣
i
∈
N
,
1
≤
j
≤
m
}
the
σ
\sigma
σ
-algebra generated by the set system coincides with the Borel set system?
topology