[分析] 實變的小lemma..

[yustarhunter (四格格)]

Lemma：

let α and μ be signed measure

If α<<μ  and  α⊥μ

then α = 0

proof：

∵α⊥μ

then there exist A,B in S  s.t  X = A∪B , A∩B = ψ , α(A) = 0 , μ(B) = 0


∵μ(B) = 0  and  α<<μ

then α(B) = 0


for all E in S

α(E) ≦ α(A∪B) = α(A) + α(B) = 0

then α(E) = 0  (這一步有問題.....α is signed measure , α(E)有可能＜E )
      ↑
     這是lemma要證的


所以我證不下去了.....有沒有人可以幫幫忙啊??

謝謝大家~~~

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推 zombiea:所以α都是負的 再考慮-α 再做一次