I've been facing the following problems:
a) Let $ X_n \rightarrow X $ and $f $ be a measurable, bounded function. Prove that $ \mathbb{E}f(X_n) \rightarrow \mathbb{E}f(X) $ (we also assume that the set of points of discontinuity of $ f $ is of measure 0)
b) Prove that $ X_n \xrightarrow{d}X \Rightarrow \mathbb{E}X \leq \liminf \mathbb{E}X_n $.
c) Let $ X_n \xrightarrow d X $ and $ \forall_n X_n$ are uniformly integrable. Prove then $ X $ is uniformly integrable.
As to a) I can't seem to be able to find a proper way to use f's measurability. All I know that pre-images of measurable sets are measurable.
In b) I obviously am going to need Fatou's lemma. I have:
$ \mathbb{E}\liminf X_n \leq \liminf \mathbb{E}X_n$. Now how can I associate the limes inferior of $ X_n $ to its 'weak' limit?