### Abstract

Let (T, A, P) be a probability space, B a P-complete sub-δ-algebra of A and X a Banach space. Let multifunction t → Γ(t), t ∈ T, have a B ⊗ B(X)-measurable graph and closed convex subsets of X for values. If x(t) ε{lunate} Γ(t) P-a.e. and y(·) ε{lunate} E_{p}^{B} x(·), then y(t) ε{lunate} Γ(t) P-a.e. Conversely, x(t) ε{lunate} F(Γ(t), y(t)) P-a.e., where F(Γ(t), y(t)) is the face of point y(t) in Γ(t). If X = R^{n}, then the same holds true if Γ(t) is Borel and convex, only. These results imply, in particular, extensions of Jensen's inequality for conditional expectations of random convex functions and provide a complete characterization of the cases when the equality holds in the extended Jensen inequality.

Original language | English |
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Pages (from-to) | 579-598 |

Number of pages | 20 |

Journal | Journal of Multivariate Analysis |

Volume | 10 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1980 |

Externally published | Yes |

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### Keywords

- conditional expectation
- face of a convex set
- Jensen's inequality
- Multifunction
- random convex set
- selector

### Cite this

*Journal of Multivariate Analysis*,

*10*(4), 579-598. https://doi.org/10.1016/0047-259X(80)90071-8