這問題出自一道習題
( Exercise 9 pp.133 W.Rudin:"Real and complex analysis"3d ed.)
Suppose that { Fn(x) } is a sequence of positive continuous functions
on I=[0,1] ( Fn≧0 ) , that μ is a positive Borel measure on I, and
that ( m : Lebesgue measure on I )
(1) lim Fn(x)→0 a.e.[m]
(2) ∫Fn(x)dm = 1 for all n
(3) lim∫g Fn(x)dm =∫g dμ for every "continuous function g(x) on I"
(All integrals are over I)
Does it follow that
μ nad m are mutually singular (μ⊥m )
我有一些例子,是符合上述條件而且
μ nad m are mutually singular (μ⊥m )
所以我"猜測"應該是μ⊥m.
但是我還沒能證出我要的結果
請各位大大給一些提示,困擾了好久.....介.介
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