Esperanza condicionada para probabilidades finitamente aditivas

• Autores: Luis Antonio Sarabia Peinador
• Localización: Trabajos de estadística e investigación operativa, ISSN 0041-0241, Vol. 33, Nº. 1, 1982, págs. 64-85
• Idioma: español
• DOI: 10.1007/bf02888704
• Títulos paralelos:
  • Conditional expectation for finitely additive probabilities
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• Resumen

  • Let (O, ?, J) be a finitely additive probabilistic space formed by any set O, an algebra of subsets ? and a finitely additive probability J. In these conditions, if F belongs to V1(O, ?, J) there exists f, element of the completion of L1(O, ?, J), such that F(E) = ?E f dJ for all E of ? 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 ß is a subalgebra of ?, for every F of V1(O, ?, J) there exists a unique element of V1(O, ?, J) which we note down by E(F/ß), conditional expectation of F given ß.

    E(F/ß) is characterized by (E(F/ß), G) = (F, G) for every G of V8(O, ß, J). Aside from this, the mapping E(./ß): V1(O, ?, J) ? V1(O, ß, J) is lineal, positive, contractive, idempotent and E(J/ß) = J. If F is of Vp(O, ?, J), p > 1, E(F/ß) is of Vp(O, ß, J