Quantum measure theory generalizes classical probability theory

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

If M is a quantum measure and \Omega is the universe set then:

1. M(\varnothing) = 0,
2. M(\Omega) = 1,
3. For any disjoint sets (measurable in the quantum sense) A, \ B and C: M(A \cup B \cup C) = M(A \cup B) + M(B \cup C) + M(A \cup C) - M(A) - M(B) - M(C)

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

M(A \cup B) = M(A) + M(B)

for all disjoint measurable sets A and B, 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