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# September 2015 Problems

1) If $X_{n}^{\prime }s$ are independent identically distributed positive
random variables does it follow that $EX_{1}X_{2}...=(EX_{1})(EX_{2})...$.
assuming that all the products and expectations exist.

2) Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be measurable and $f(x+y)-f(x)$ be continuous for all $y$ in some set of positive measure. Show that $f$ is continuous. Show that measurability cannot be dropped. If $f(x+y)-f(x)$ be continuous for all $y$ in some set with a limit point can we conclude that $f$ is continuous? Does there exist a set $A$ of measure $0$ such that if $f(x+y)-f(x)$ is continuous for all $y$ in $A$ then If $f$ is necessarily continuous? If $A$ is at most countable show that there exist $f$ such that $f(x+y)-f(x)$ is continous for all $x$ in $A$ but $f$ is not continuous. If $f:\mathbb{R}\rightarrow \mathbb{C}$ is continuous and $f(x+y)-f(x)$ is an entire function for all $y$ in some set with a limit point can we conclude that $f$ is entire?

3) Let $P$ and $Q$ be projections onto closed subspaces $M$ and $N$ of a
Hilbert space $H.$ Find a necessary and sufficient condition on $M$ and $N$
for $PQ$ to be a projection.