Let $\left(X_{n}\right)$ be a sequence of non-negative random variables on a common probability space with $\mathbb{E} X_{n} \leqslant 1$, such that $X_{n} \rightarrow 0$ almost surely. Determine which of the following statements are necessarily true, justifying your answers carefully: (a) $\mathbb{P}\left(X_{n} \geqslant 1\right) \rightarrow 0$ as $n \rightarrow \infty$; (b) $\mathbb{E} X_{n} \rightarrow 0$ as $n \rightarrow \infty$; (c) $\mathbb{E}\left(\sin \left(X_{n}\right)\right) \rightarrow 0$ as $n \rightarrow \infty$; (d) $\mathbb{E}\left(\sqrt{X_{n}}\right) \rightarrow 0$ as $n \rightarrow \infty$.

[Standard limit theorems for integrals, and results about uniform integrability, may be used without proof provided that they are clearly stated.]