I'd like to know if these claims are true or false before attempting to prove them.

Question

let $(X_n)_{n \geq0}$ be a uniformly integrable martingale

does it converge in $L^1$ ? $L^2$ ? almost surely ?

I don't want a proof, I just want to know which claims are true so I can attempt proving them myself.

Answer 1

A martingale is uniformly integrable if and only if it converges in $L^1$.

A uniformly integrable martingale converges almost surely.

For convergence in $L^2$ we need to know a bit more. (for example the martingale being bounded in $L^2$ is sufficient)