Probability without second axiom (unit measure) I'm working with functions (namely, representing incoherent degrees of belief) which resemble probabilities, but are actually, say, _quasi-probabilites_ : * their values on atomic events (here: atomic propositions) are in $[0;1]$, but * they need not to sum up to $1$. For example, if we have belief space $B=\\{\phi_1$, $\phi_2\\}$ and some credence function $c$, then it may be the case that $c(\\{\phi_1\\})=0.5$ and $c(\\{\phi_2\\})=0.7$, so obviously $c(B)\neq1$. Nevertheless, it's always $c(\\{\phi_i\\})\in[0;1]$. This violation of probability laws creates many theoretical problems, so I'm in need of some proper theoretical framework. But I don't want to reinvent the wheel. I wonder if there was any attempt to formulate alternative probability theory without the axiom of unit measure, so that not necessarily $P(\Omega)=1$? Edit: In particular, I need something like conditional probability and Bayes' theorem.

Probability theory absent unit measure is developed by R. Christensen and T. Reichert: "Unit measure violations in pattern recognition: ambiguity and irrelevancy" Pattern Recognition, 8, No. 4 1976.