A random variable (r.v.) is a (measurable) fucntion from probability space $\Omega$ to R. In our applied problem, the best model would be an extended "r.v." from $\Omega$ to $R\cup{\infty}$. For such "r.v." the cumulative distribution function can be defined naturally, it will be a rightcontinuous nondecreasing function with $F(\infty)=1$ but with $F(\infty)$ not nessesary $0$. Expectation does not exists if $P(X=\infty) > 0$ but conditional expectation $E[X  X > \infty]$ makes sence. Next step is to define different types of convergence (in probability, in distribution, etc.). The question is: Is this or similar "probability theory" known? Maybe, it can be derived as a corollary of some more general theory, etc.? I would be happy to develop it myself, but afraid to "reopen" a wellknown theory.

The definition you gave is a special case of the definition I'm used to, where the target space is an arbitrary measurable space or topological space instead of the real line. Your extended version fits just fine into that framework. See: http://en.wikipedia.org/wiki/Random_variable#Formal_definition 


While you are at it, you could allow the value $+\infty$ as well... The resulting object is a random variable from a probability space $(\Omega,\mathcal{F},P)$ to a bona fide measurable space $(E,\mathcal{E})$, in this case $E=\mathbb{R}\cup\{\infty,+\infty\}$ and $\mathcal{E}$ the Borel $\sigma$algebra of $E$ endowed with its usual topology (roughly speaking, this is equivalent to seeing $E$ as the closed interval $[0,1]$ endowed with its subspace topology). A description of $\mathcal{E}$ is that $A\subset E$ is in $\mathcal{E}$ iff there exists $B\in\mathcal{B}(\mathbb{R})$ and $I\subset\{\infty,+\infty\}$ such that $A=B\cup I$. So you are squarely in probability theory in the most standard sense of the term. Rereading your post, I feel I should mention that $E(XX\in\mathbb{R})$ need not be well defined, for the same reason that a real valued random variable need not be integrable. 

