Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
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.
Typology: Lecture notes
1 / 1
This page cannot be seen from the preview
Don't miss anything!
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)
