1. Let and be random variables such that all have the same distribution. If the common distribution has finite mean show that a.s. Prove that the assumption on finiteness of the mean cannot be dropped.
2. Prove that a function from one metric space to another is uniformly continuous if and only if implies where .
3. [ Contributed by Manjunath Krishnapur] Let be a probability space. Suppose is an increasing sequence of sigma algebras on contained in and is a decreasing sequence of sigma algebras on contained in such that is trivial. Let be a random variable on which is measurable w.r.t. for each . Does it follow that is measurable w.r.t. the completion of the sigma algebra generated by all the ?
Note:- A sigma algebra is trivial w.r.t. a probability measure if every set in it has probability or and is the sigma algebra generated by