※ 引述《TrySoHard (夢永遠都只是夢)》之銘言：
: 1. Does there exist an infinite σ-algebra that has only countably
: many element?
Let A be an infinite σ-algebra on a space X. Pick an arbitrary point x_1
in X. Since A is an algebra, both
A = { B in A｜B contains x_1}, A = { B in A｜x_1 not in B}
1 0
are nonempty. Since A is infinite, there exist x_2 != x_1 such that either
(i) there is some set B in A_1 that contains x_2 and another B' in A_1
that does not contain x_2;
(ii) there is some set B in A_0 that contains x_2 and another B' in A_0
that does not contain x_2.
Consequently, since A is an algebra, the following four collections are
nonempty:
A = { B｜x_1 in B, x_2 in B }, A = { B ｜x_1 in B, x_2 not in B },
11 10
A = { B｜x_1 not in B, x_2 in B }, A = { B｜x_1 not in B, x_2 not in B }.
01 00
Proceed inductively. Assume that there exists n distint point x_1, ...,x_n
such that the corresponding 2^n collections
A (i = 0 or 1, j = 1, ...,n )
i_1 ... i_n j
are nonempty. Since A is infinite, at least one of the collections contains
two distinct sets. Denote the two sets by B* and C*. Now that B* and C* are
different subsets of X, there exists x_{n+1}, different from x_1, ..., x_n,
such that x_{n+1} is contained in one of B* and C* and not in the other.
Say x_{n+1} in B* and x_{n+1} not in C*.
Since A is an algebra, after taking intersection of B* with X-C*, we know
A = { B in A ｜ x_1 in B }
00 ...01 00 ...0
╰─n─╯ ╰─n─╯
is nonempty. After takeing intersection of a set in C with a set in
A_{00...0}(length=n), we know
A = { B in A ｜ x_1 in B }
00...00 00 ...0
╰─n+1╯ ╰─n─╯
is nonempty. Therefore, by taking union of an element in A_{i_1 i_2 ...i_n}
with an element from one of the previous two collections, we know both
A and A
1_1,i_2,...,i_n,1 1_1,i_2,...,i_n,0
are nonempty. So we are done with induction. We have the following:
There exits a sequence of distinct points x_1, x_2, ... such that for each n
and for each (i_1,i_2,...,i_n) in (Z_2)^n the corresponding collection
A is nonempty.
i_1 i_2 ... i_n
In particular, the following kind of collection is nonempty:
i th
A ↙ ,
0 ...0 1 0 ...0
for arbitrary finite length n and arbitrary position i = 1,2,...n. Finally,
since A is a σ-algebra, taking countable unions of sets from collections
of the previous form makes
A
i_1 i_2 i_3 ....
nonempty, for each infinite sequence (i_1,i_2,i_3,...). This done, the
cardinality of A is at least 2^│|N│=│|R│.
: 2. Let E be a measurable subset of [0,1]. Let A={ x^1/2 | x is in E }.
: Is A a measurable subset of [0,1]?
: 先感謝各位解答了
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◆ From: 114.32.4.99
※ 編輯: ppia 來自: 114.32.4.99 (11/23 11:56)