# Relationship between Abstract Algebra and Probability Theory

Is there a relationship between abstract algebra and probability theory? I ask this because of the following laws:

Axiom of Countable Additivity: If $A_1, A_2, \dots \in \mathcal{B}$ (where $\mathcal{B}$ is a $\sigma$-algebra) are pairwise disjoint then $$P \left(\bigcup_{i=1}^{\infty} A_i \right) = \sum_{i=1}^{\infty} P(A_i)$$

and

Axiom of Finite Additivity: $P(A \cup B) = P(A)+P(B)$ for $A,B \in \mathcal{B}$ where $A$ and $B$ are disjoint.

Note that the first implies the second. Also the second implies the first if we assume the axiom of continuity:

Axiom of Continuity. If $A_n \downarrow \emptyset$ then $P(A_n) \to 0$.

So it seems that we can treat these probabilities are homomorphisms. So is there any relationship between abstract algebra and probability theory?

• Whoa! Homomorphisms have to respect the operation for all pairs of arguments, not just the disjoint ones, so I can't see treating probabilities as homomorphisms. – Gerry Myerson Dec 25 '11 at 1:53
• You can think of the connection between probability and algebra in terms of the sigma algebra of sets from your sample space. Perhaps see en.wikipedia.org/wiki/Ring_of_sets – mathmath8128 Dec 25 '11 at 3:04
• Gerry Myerson is right: $P(A\cup B) = P(A)+P(B)$ if $A$ and $B$ are mutually exclusive, but not generally if they're not. This doesn't mean there's no connection, but maybe that it's not a really simple one. – Michael Hardy Dec 25 '11 at 7:18
• See, for instance, "Partial probability and Kleene Logic: arxiv-web3.library.cornell.edu/abs/1310.6172 – user122271 Jan 18 '14 at 18:32