Multifunctions of faces for conditional expectations of selectors and Jensen's inequality

A. Kozek*, Z. Suchanecki

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


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} EpB 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 = Rn, 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 languageEnglish
Pages (from-to)579-598
Number of pages20
JournalJournal of Multivariate Analysis
Issue number4
Publication statusPublished - 1980
Externally publishedYes


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


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