Sum of Discrete Random Variables

Theorem
Let $X$ and $Y$ be discrete random variables on the probability space $\struct {\Omega, \Sigma, \Pr}$. Let $U: \Omega \to \R$ be defined as:
 * $\forall \omega \in \Omega: \map U \omega = \map X \omega + \map Y \omega$

Then $U$ is also a discrete random variable on $\struct {\Omega, \Sigma, \Pr}$.

Proof
To show that $U$ is discrete random variable on $\struct {\Omega, \Sigma, \Pr}$, we need to show that:


 * $(1): \quad$ The image of $U$ is a countable subset of $\R$;
 * $(2): \quad \forall x \in \R: \set {\omega \in \Omega: \map U \omega = x} \in \Sigma$.

First we consider any $U_u = \set {\omega \in \Omega: \map U \omega = u}$ such that $U_u \ne \O$.

We have that $U_u = \set {\omega \in \Omega: \map X \omega + \map Y \omega = u}$.

Consider any $\omega \in U_u$.

Then:
 * $\omega \in X_x \cap Y_x$

where:
 * $X_x = \set{\omega \in \Omega: \map X \omega = x}, Y_x = \set {\omega \in \Omega: \map Y \omega = u - x}$

Because $X$ and $Y$ are discrete random variables, both $X_x \in \Sigma$ and $Y_x \in \Sigma$.

As $\struct {\Omega, \Sigma, \Pr}$ is a probability space, then $X_x \cap Y_x \in \Sigma$.

Now note that:
 * $\displaystyle U_u = \bigcup_{x \mathop \in \R} \paren {X_x \cap Y_x}$

That is, it is the union of all such intersections of sets whose discrete random variables add up to $u$.

As $X_x$ is a countable set it follows that $U_u$ is a countable union of countable sets.

From Countable Union of Countable Sets is Countable it follows that $X_x$ is a countable set.

And, by dint of $\struct {\Omega, \Sigma, \Pr}$ being a probability space, $U_u \in \Sigma$.

Thus $U$ is a discrete random variables on $\struct {\Omega, \Sigma, \Pr}$.