There is a quantum measure theory (an extension to the mathematical discipline called “measure theory”) that goes as follows:

If is a quantum measure and is the universe set then:

1. ,

2. ,

3. For any disjoint sets (measurable in the quantum sense) and :

Notice that, if and are disjoint sets then, in some quantum experiments, cannot be always measured from the measurements of each isolated piece and as is usually considered in the classical measure theory. In these cases, we must compute a specific measure for the set . Naturally, if

for all disjoint measurable sets and , then the usual probability measure emerges, but it is not the case in quantum experiments. The axiom 3. is called grade-2 additivity

There is a connection between M and the wave function. For more on this, just google it: “quantum measure theory”.

Best,

Alexandre Patriota