August | 2015 | Lecture Notes, Exercises etc

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 $\mathcal{F}_{n}\cup \mathcal{G}_{n}$