# 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