Note that as stated, the 'rule,' mentioned above, has multiple interpretations.
$$(A) P(A\vee B)=P(A)+P(B)−P(AB)$$ as stated an axiom of Kolmogorov's system.
Unless you interpret $$P(AB)$$ as $P(A\cap B)$ as set theoretic intersection, which is how its, or was, traditionally done.
Where $A\cap B$ is the event $E;E \in F$.
$F$ being the Bool-ean algebra of events.
Where the event $E$ , is, denoted by a set of atomic events:
$\Omega_{i};\Omega_{i} \in \Omega$, generally if not always, mutually exclusive, where $\Omega$ is the sample space .
Or in other words, the union of singleton sets: $\{\Omega_{i}\} \in F, \{\}$ ,denoting said atomic events, which are often measurable in finite cases, and thus $\in F$, the 'Boolean or sigma algebra of measurable events'. This being is often, the union of the singleton sets (translate , dis-junction, of 'mutually exclusive 'atomic event's) , that are in the common intersection of the sets $E_{A}\,,E_{B}$ .
These are the union of singleton sets in ${\Omega_{i}}\in F$ (translate , dis-junction, of 'mutually exclusive 'atomic event's; $\Omega_{i}\in \Omega$), ' which denotes the events $A , B$ respectively.
So at the end of the day, one never consider composite events really, in basic Kol-mogorov's calculus, that are not mutually exclusive.
By which I mean they are always reducible to dis-junctions of mutually exclusive events. Whilst some events in $F$ may not be mutually exclusive, what those events consist in.
That is, the elements of the sets denoting each event individually, are generally if not always mutually exclusive, are, so there are no compositions (intersection of union )or events consisting of non mutually exclusive atomic events, or other partitions.
Unless you count $\emptyset$ .
Otherwise union of singleton sets, or atomic events, or countable partitions of them (unions) in the infinite cases, are generally always be analyzed set theoretically via as the disjoint union, and set co of mutually exclusive events, and set complementation so will $A\cap B$.
Although, the probability calculus was in a sense, extended later, by both Kol-mogorov and others, as it stands, the three axioms of probability $(1)$ , $(2)$ and$ (3)$ do not contain $$(A)$$, at least not explicitly.
Everything can be done by summing up only mutually exclusive events. Its just a quicker theoretical tool.
Perhaps the real, difference is, at not so much at the level of axioms, the nature of the probability space. However, the underlying model theory, logic, measure theory and sets, on the one hand, versus, functions, vector spaces, and inner-products, and what is meant by disjoint, and complementary and closure under unions of certain forms
See Chapter 5 of 'Foundations of Measurement Volume 1: Suppes Krantz, Luce et al for more on the distinction in quantum mechanics.
and the that quantum probability may be considered non-commutative, and may see a distinction in the logic between the events(1) and (2):
$(1)P([A\cup B ]\vee [B\cup C])$ and $(2)PR(A\vee B \vee C)$ .
Where $A\cap B$ and $B \cup C=\emptyset $.
But, in the conventional system (such as kolmogorov), $P([A \vee B ]\land [B\vee C])$, is interpreted, set-theoretically as:$ P([A\cup B ]\cap [B\cup C])=P({B})$ for example.
Where $A , B C$ are the atomic, mutually exclusive and exhaustive events.
Thus, the probability calculus, really only contains $$(2)$$ interpreted in terms of set theoretic intersection, comple-mentation and unions.
Although generally intersection is not required given specification of the unit.
$A\cap B$ will generally be an event $\in \mathbb{F}\subseteq=2^{\Omega}$ , in algebra of events $F$. That is a set of mutually exclusive elementary outcomes ${D,C...}$ or a singleton set,${D}$\in $F$, $D$\in $\Omega$
an atomic event, singleton,or union of said mutually exhaustive events, generally, in the algebra of events, as will every other event, generally
$A\cap B=$ ,sing set theoretic comple-mentation.
Moreover, $P(AB)$ is not interpreted as some kind of product or multiplicative event it is always either a union an atomic events, or the empty event or the unit event at the end of the day, when one using set theoretic intersection for a singular probability space.
There are no distinct product events in the algebra over and above these.
$$(1)P(A \vee B)=P(A)+P(B)−P(A\land B)$$.
$$(2)P(A\cup B)=P(A)+P(B)−P(A\cap B)$$.
It has been debated whether Kolmorogov's original axiom set is incompatible with quantum mechanics.
Maybe its official extension to measure theory, to joint algebras, product sets, what have you, to definite laws of large numbers, independence and so forth. But the original axioms say nothing about.
A lot of the counter-examples I have sen in other articles, have nothing to with Kolmogorov's official formulation which says nothing about products, independence,Bayes' axiom or the multiplicative /product'axioms'.
Which are put forward as mere definitions, and by definition this means that:
$P(A|B)$ is a word that merely denotes $\frac{P(AB)}{P(B)} .
Where $P(A|B)=\frac{P(AB)}{P(B)}$ then says that the ratio of two probabilities $\frac{P(AB)}{P(B)} =\frac{P(AB)}{P(B)}$ equals the the ratio of the same two probabilities (ie nothing substantive).
This is because, conditional probabilities, are not explicitly part of the axiomatic structure derived from the measure theoretic interpretation of probability,if you can call it that.
Quite, unlike the addiv-itiy of probabilities over disjoint unions, which was 'derived' from the addivity of under-lying inter-pretation (measure).
Nor is, $P(A|B)$, the probability of a conditional even, at least in the canonical calculus of probability. Nor, are the definitions of independence and Bayes rule (definition really).
These, independence, Bayes rule, conditional probabilities etc, are not officially not axioms of Kol-mogorov's system. hey are definitions!
There may have been reasons,why Kolmorogov, did not officially publish his results earlier, which were published instead at about the same time that Quantum probability-logic of Von Neumann /Birk-hoff etc. These works, have the same official vintage.
Kol-mogorov, may have been aware of some of the issues. Perhaps, part of the reason for his his being tentative about considering or formalizing the product definition, Bayes rule, and independence, as axioms of probability.
Whilst, Kol-mogorov never did officially include these as axiom but only definitions, the probability calculus was extended later, by both himself and others to accommodate Bayes rule to put it on a more formal axiomatic like standing.
Nonetheless, as it stands, the three axioms of probability still do not contain contain it (Baye's Rule) or independence, and thus the 'product rule' ,$(2)$, or the rule of 'total probability', $(1)$ , as below, as axioms:
$$(1)P(A \\vee B)=P(A)+P(B)−P(A\land B)$$ .
$$(2)P(A\cup B)=P(A)+P(B)−P(A\cap B)$$.
The only official use of is $(2)$ , as an axiom, where roughly :
$P(A\cup B)= P(A\\B)+P(B)$. But this makes no use of the notions of definitions of conditional probabilities, multip-licativity of probabilities as in the product rule, or independence.
Axioms: Where a Probability space is a triple :
$$\langle \Omega,\mathbb{F}, P\rangle$$.
Where, $\Omega$ is the sample space, the set of mutually exclusive and exhaustive atomic events, ,often called single-tons events,$\in$ the algebra $\mathbb{F} $.
Where, $\mathbb{F} \subseteq \mathcal{P(\Omega)}=2^{\Omega}$, is an Bool-ean algebra events, a set of measurable subsets, of $\Omega$, closed under the unit event (certain event:$\Omega$, complementation, and countable union.
$$(1)P(E)\in \mathbb{F} ,P(E)\geq 0\quad \forall E\in F$$
$$(2) P(\Omega)=1$$.
$$(3)P (⋃^{i=\infty}_{i=1}E_{i}) = $$.
$$\lim_{n \to \infty} ∑^{i=n} _{i=1}P(E_{i})$$.
Where: $\forall A_{i}$ are mutually exclusive. That is, pairwise disjoint.
A countable sequence of , pairwise-disjoint sets, which requires an axiom of continuity, and sometimes is formalized as fourth axiom distinct from finite additivity.
If you read Kol-mogorov's main work, he states many tentative axioms such as axiom $4|5$ the frequency principle which he decided to reject in the end .
Whilst Kol-morogov is often considered to be a frequentist in certain circles, in the end he rejected the formal frequent-ism of Von Mises/Reichen-bach, as did Von -Neumman in Quantum- Mechanics .Kol-morogov may have been still a frequentist at heart. He said explicitly that his model did not have a great deal to say, or was neutral about the world, and that frequency data, is how you obtain information, but formally speaking he rejected it in the end.
Any countable sequence of disjoint sets:
with mutually exclusive events ,roughly:
$E_{1} ,\, E_{2} \text{..}\in F$ , satisfies:
.
$$P(⋃_{i=1}^{\infty}E_{i})= \, \sum_{i=1}^{\infty }P(E_{i}\,)$$.
