Rigorous definition of sum of two random variables? I've looked everywhere, but I seem unable to find a rigorous definition (in the set-theoretical sense of a random variable being a mapping from a sample space to the real number line) of the sum of two random variables (X + Y). 

How do you rigorously define the sum of two random variables X + Y?
Given random variables $X, Y : \Omega\to\mathbb R$ defined on the same probability space $\Omega$, the definition of $X+Y$ is simply the pointwise sum:
$$(X+Y)(\omega) = X(\omega)+Y(\omega)$$
If $X$ and $Y$ are defined on different probability spaces, $X:\Omega_1 \to \mathbb R$ and $Y:\Omega_2 \to \mathbb R$, then $X+Y$ is undefined. However, in this case we can define a new probability space $\Omega = \Omega_1 \times \Omega_2$ and random variables $X_1, Y_1$ on $\Omega$ by
\begin{align*}
X_1(\omega_1,\omega_2) &amp;= X(\omega_1)\\
Y_1(\omega_1,\omega_2) &amp;= Y(\omega_2)
\end{align*}
in which case $X_1$ has the same distribution as $X$, $Y_1$ has the same distribution as $Y$, and $X_1$ and $Y_1$ are independent. In this case, the sum $X_1+Y_1$, which is defined pointwise as above, may be considered as a replacement for the non-existent sum $X+Y$, as long as independence between the two summands is what we were looking for.
It actually defined as an analogue of the product of the polynomial with $X, Y$ independent and P was defined in double dimension

$$(P(x + y) = t) =  \sum_{I = 0}^t f_x(I)g_x(t - I)$$