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Two problems related to the convergence of functions with respect to measures in probability theory. The first problem deals with the lower semicontinuity of a bounded function and the convergence of measures, while the second problem discusses the continuity of a bounded function and the convergence of measures. Hints are provided for each problem.
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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.
that wheneverQ2. If^ f^ (x ) is bounded and continuous at every point ofμ^ Ac^ and^ μ(A) = 0 then show n ⇒^ μ n^ lim→∞
f (x)dμn(x) =
f (x)dμ(x)
