Let (Ω, θ, J) be a finitely additive probabilistic space formed by any set Ω, an algebra of subsets θ and a finitely additive probability J. In these conditions, if F belongs to V1(Ω, θ, J) there exists f, element of the completion of L1(Ω, θ, J), such that F(E) = ∫E f dJ for all E of θ and conversely.
The integral representation gives sense to the following result, which is the objective of this paper, in terms of the point function: if β is a subalgebra of θ, for every F of V1(Ω, θ, J) there exists a unique element of V1(Ω, θ, J) which we note down by E(F/β), conditional expectation of F given β.
E(F/β) is characterized by (E(F/β), G) = (F, G) for every G of V∞(Ω, β, J). Aside from this, the mapping E(./β): V1(Ω, θ, J) → V1(Ω, β, J) is lineal, positive, contractive, idempotent and E(J/β) = J. If F is of Vp(Ω, θ, J), p > 1, E(F/β) is of Vp(Ω, β, J).
@article{urn:eudml:doc:40677,
title = {Esperanza condicionada para probabilidades finitamente aditivas.},
journal = {Trabajos de Estad\'\i stica e Investigaci\'on Operativa},
volume = {33},
year = {1982},
pages = {64-85},
zbl = {0513.60006},
mrnumber = {MR0697208},
language = {es},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:40677}
}
Sarabia, Luis A. Esperanza condicionada para probabilidades finitamente aditivas.. Trabajos de Estadística e Investigación Operativa, Tome 33 (1982) pp. 64-85. http://gdmltest.u-ga.fr/item/urn:eudml:doc:40677/