OK. I wanted to leave even that part open in case there are particular definitions of Sum and Prod for which the measure distributes inside the Sum or the Prod. But initially my first guess is that Sum is defined as traditional addition of numbers, and Prod is defined as traditional multiplication of numbers, and the P_i are sets.
For example, as I understand measure theory, in order that m(P1 union P2)=m(P1)+m(P2), there must be the restriction that P1 intersect P2 = empty set. In other words, P1 and P2 must be disjoint. In all the books I've looked at, this seems to be given as an axiom that's not proven. Yet, I wonder if there is a similar or perhaps dual requirement for Prod?
Also, my goal is to be able to somehow put a measure on the space of propositions so that disjunction and conjunction get translated to addition and multiplication of measures on propositions or on sets of propositions. Can one get from the more traditional treatments of measures on sets to getting measures on propositions by letting the number of elements in a set go to 1 element? Then propositions could be labeled synonymously with its set. Would this turn unions and intersections into disjunction and conjunction? Any help would be very much appreciated.