A random variable (r.v.) is a (measurable) fucntion from probability space $\Omega$ to $\mathbb{R}$. In our applied problem, the best model would be an extended "r.v." from $\Omega$ to $\mathbb{R}\cup\{-\infty\}$. For such "r.v." the cumulative distribution function can be defined naturally, it will be a right-continuous 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 $E[X \mid X > -\infty]$ 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 well-known theory.
[Math] A “random variable” with infinite value
pr.probability
Best Answer
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 $E(X|X\in\mathbb{R})$ need not be well defined, for the same reason that a real valued random variable need not be integrable.