Suppose g_n is a sequence of positive continuous functions on I = [0,1],
u is a positive Borel measure on I, m is the Lebesgue measure, and that
1. lim g_n(x) = 0 a.e. [m]
n-->∞
2. ∫g_n dm = 1
I
3. lim ∫fg_n dm = ∫f du for every continuous function f.
n-->∞ I I
Does it follow that u ⊥ m ?
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※ 編輯: hips 來自: 220.135.112.28 (12/15 23:48)