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.

Advertisements