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 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 $E[X | X > -\infty]$ 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.

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.