Finite probabilistic structures

Consideration of the field of events in the theory of probability gives rise to the notion of the field of events $\mathscr F(B)$ consisting of a set of subsets of some set $B$. On the field $\mathscr F(B)$, two algebraic structures are naturally defined. These are the Boolean algebra $\mathscr A(\mathscr F(B))$ with the operations of union, intersection and complement, and the lattice $L(\mathscr F(B))$, where the order is defined according to inclusion of the sets of $\mathscr F(B)$. In this paper, we consider one more algebraic structure on $\mathscr F(B)$ and the abstract variant of this structure, the so-called probabilistic structure, which is closely related to properties of the measure on $\mathscr F(B)$.