1) If are independent identically distributed positive

random variables does it follow that .

assuming that all the products and expectations exist.

2) Let be measurable and be continuous for all in some set of positive measure. Show that is continuous. Show that measurability cannot be dropped. If be continuous for all in some set with a limit point can we conclude that is continuous? Does there exist a set of measure such that if is continuous for all in then If is necessarily continuous? If is at most countable show that there exist such that is continous for all in but is not continuous. If is continuous and is an entire function for all in some set with a limit point can we conclude that is entire?

3) Let and be projections onto closed subspaces and of a

Hilbert space Find a necessary and sufficient condition on and

for to be a projection.