Esperanza condicionada para probabilidades finitamente aditivas

Let (Ω, θ,J) be a finitely additive probabilistic space formed by any set Ω, an algebra of subsets θ and a finitely additive probabilityJ. In these conditions ifF belongs toV ¹ (Ω, θ,J) there existsf, element of the completion ofL ¹ (Ω, θ,J), such thatF(E)=∫ _E fdJ for allE of θ and conversely. This integral representation gives sense to the following result, which is the objetive of this paper, in terms of the, point function: If β is a subalgebra of θ, for everyF ofV ¹ (Ω, θ,J) there exists a unique element ofV ¹ (Ω, β,J) which we note down byE(F/gb), conditional expetation ofF given β. E(F/β) is characterized by (E(F/β),G)=(F, G) for every ofV ^∞(Ω,β, J). Aside from this, the mappingE(./β):V ¹ (Ω, θ,J)→V ¹ (Ω, β,J) is linear, positive, contractive, idempotent andE(J/β)=J. IfF is ofV ^p (Ω, θ,J),p>1,E(F/β) is ofV ^p (Ω, β,J).