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Probability, Limit Theorems

Problem set 3. Due Oct 10, 2002

Q1. If f(x) is a bounded lower semicontinuous function on R and µn ⇒ µ show that∫ f(x)dµ(x) ≤ lim inf

n→∞

∫ f(x)dµn(x)

Hint: Write f(x) = lim ↑ fn(x) an increasing limit of bounded continuous functions.

Q2. If f(x) is bounded and continuous at every point of Ac and µ(A) = 0 then show that whenever µn ⇒ µ

lim n→∞

∫ f(x)dµn(x) =

∫ f(x)dµ(x)

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