Let $X$ be a random variable with space $\mathcal{D}$. For $D \subset \mathcal{D}$, recall that the probability induced by $X$ is $P_{X}(D)=P[\{c: X(c) \in D\}] .$ Show that $P_{X}(D)$ is a probability by showing the following:
(a) $P_{X}(\mathcal{D})=1$.
(b) $P_{X}(D) \geq 0$.
(c) For a sequence of sets $\left\{D_{n}\right\}$ in $\mathcal{D}$, show that
$$
\left\{c: X(c) \in \cup_{n} D_{n}\right\}=\cup_{n}\left\{c: X(c) \in D_{n}\right\}
$$
(d) Use part (c) to show that if $\left\{D_{n}\right\}$ is sequence of mutually exclusive events, then
$$
P_{X}\left(\cup_{n=1}^{\infty} D_{n}\right)=\sum_{n=1}^{\infty} P_{X}\left(D_{n}\right)
$$