I wonder whether the law of excluded middle not being available in intuitionistic logic might provide an obstacle when working with an axiomatization of probability theory using the Kolmogorov axioms. More precisely, if we cannot always know whether $x \in A$ or $x \notin A$ for arbitrary $x$, how can we know that $P(A) + P(\Omega \backslash A) = 1$?
So, can the Kolomogorov axioms be reasonably used in intuitionistic logic, or is there a better axiomatization of probability that works well with intuitionistic logic?