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?