※ 引述《ericpony (活死人)》之銘言：
: 如題, 請版上高手提供證明思路或反例, 謝謝!
: (原題目是 Show that if u(.) is a finite measure, then there
: cannot be uncountably many disjoint sets A such that u(A)>0.)
Suppose there exists uncountably many disjoint sets {Ai | i in I}
u(Ai) > 0 for each i in I. I uncountable index set
Devide I into such sub-collection: In = {i | u(Ai) > 1/n}
Then I = union of all In
If there's an Ik that has infinite elements,
then let E = union A_i
i in Ik
We'll have u(E) = sum u(A_i) = ∞, a contradiction.
Hence each Ik is finite => I = union Ik is at most countable.
Still contradicts that I is uncountable.
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Problem: Let {x_n} be a convergent real sequence.
Which of the following is true?
(a) {x_n} is Cauchy. (b) {x_n} is Riemann.
(c) {x_n} is Galois. (d) {x_n} is Abel.
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◆ From: 111.249.203.156
※ 編輯: chy1010 來自: 111.249.203.156 (11/14 01:38)