推 ericakk :感謝你 *^.^* 11/01 10:37
※ 引述《ericakk (穎 我好思念你...)》之銘言：
: 麻煩各位高手了～感激不盡^^
: If (Χ,Μ,μ) is a measure space and {E_j}∞ in Μ ,
: 1
: ∞
: μ(∪E_j) ＜∞ .
: 1
: Prove that:
: μ(lim inf E_j) ≦ lim inf μ(E_j) ≦ lim sup μ (E_j) ≦ μ (lim sup E_j)
For the first inequality:
Consider {1_{E_i}}, the indicator functions of measurable sets E_i's.
Observe that liminf 1_{E_i} = 1_{liminf E_i}
In fact, liminf 1_{E_i}(x)=1 iff x sits in liminf E_i.
Otherwise, liminf 1_{E_i}(x)=0
Now apply Fatou's lemma,
μ(liminf E_i)=∫1_{liminf E_i}dμ
∫liminf 1_{E_i}dμ≦liminf∫1_{E_i}dμ=liminf μ(E_i)
For the third inequality:
Let S = ∪E_j, j runs over 1,2,... By assumption, μ(S) is finite.
Put F_i=S-E_i. Then 1_{F_i}=1_{S}-1_{E_i}.
Applying Fatou's lemma on 1_{F_i} again. You'll get the result.
Notice that the finiteness of μ(S) is only used in this part.
The second one is trivial by definition.
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※ 編輯: yusd24 來自: 140.112.51.100 (11/01 09:03)