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August 2015 Problems

1. Let X and Y be random variables such that X,Y,X+Y,X-Y all have the same distribution. If the common distribution has finite mean show that X=0 a.s.  Prove that the assumption on finiteness of the mean cannot be dropped.

2. Prove that a function f from one metric space to another is uniformly continuous if and only if d(A,B)=0 implies d(f(A),f(B))=0, where d(A,B)=\inf\left\{d(x,y):x\in A,y\in B\right\}.

3. [ Contributed by Manjunath Krishnapur] Let (\Omega ,\mathcal{F},P) be a probability space. Suppose \{\mathcal{F}_{n}\} is an increasing sequence of sigma algebras on \Omega contained in \mathcal{F} and \{\mathcal{G}_{n}\} is a decreasing sequence of sigma algebras on \Omega contained in \mathcal{F} such that \bigcap\limits_{n}\mathcal{G}_{n} is trivial. Let X be a random variable on (\Omega ,\mathcal{F},P) which is measurable w.r.t. \sigma\{ \mathcal{F}_{n},\mathcal{G}_{n}\} for each n. Does it follow that X is measurable w.r.t. the completion of the sigma algebra generated by all the \mathcal{F}_{n}^{\prime }?

Note:- A sigma algebra is trivial w.r.t. a probability measure P if every set in it has probability 0 or 1 and  \sigma \{\mathcal{F}_{n},\mathcal{G}_{n}\}  is the sigma algebra generated by \mathcal{F}_{n}\cup \mathcal{G}_{n}

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