※ 引述《k6416337 (とある煞氣の光希)》之銘言:
(略)
: → cgkm :limsup E_k 的測度為零的證明: http://0rz.tw/5ztzm 11/15 18:00
: → cgkm :然後 liminf E_k 包含於 limsup E_k 11/15 18:03
: → k6416337 :我老師的解答寫得比這還長很多 11/15 18:18
兩個超短的證明:
Proof 1.
μ( ∪ E ) ≦ Σ μ(E ) → 0.
k≧n k k≧n k
and the term on the left converges to μ(lim sup E ). □
k
Proof 2. Let f(x) = Σ 1 (x) = the number of E containing x.
k E k
k
Then ∫ f = Σ μ(E ) < ∞ by the monotone convergence theorem,
k k
so f = ∞ (i.e. x occurs infinitely often) on a set of measure 0. □
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