What is the formalism that allows Random Variables to be treated algebraically like real or complex numbers? We all know that if we have a variable x, then there is a meaning to - for example - 
$$y=e^x$$.  
And we all know how to manipulate that algebraically and to do calculus.  For example, if 
$$y_1=e^{x_1}$$
and 
$$y_2=e^{x_2}$$
then 
$$y_1y_2 = e^{x_1+x_2}$$
But random variables are a very different beast. Ultimately we all know intuitively what we mean when we draw a PDF or, more accurately, the CDF which is itself an algebraic function mapping $[-\infty, \infty]$ to $[0,1]$ as part of that defines a random variable.
But what is the actual math that makes that definition of an RV actually act algebraically, so that, as above, we can happily write things like 
$$Y_1=e^{X_1}$$
and 
$$Y_2=e^{X_2}$$
then 
$$Y_1Y_2 = e^{X_1+X_2}$$
where $X_1$ and $X_2$ are random variables - and get away with it???

Formally $X_i$ are real valued (measurable) functions i.e $X_i\colon \Omega\to \mathbb{R}$ for $i=1,2$ where $\Omega$ is the sample space. So 
$$
Y_i=e^{X_i}
$$
for $i=1,2$ as an equality of functions means that
$$
Y_i(\omega)=e^{X_i(\omega)}
$$
for all $\omega\in \Omega$. Hence since $X_i(\omega)$ is a real number it follows using the first part of your question that
$$
Y_1(\omega)Y_2(\omega)=e^{X_1(\omega)}e^{X_2(\omega)}=e^{X_1(\omega)+X_2(\omega)}
$$
for all $\omega$ whence as functions
$$
Y_1Y_2=e^{X_1+X_2}.
$$
