2023/05/31 by Kazimierz Musiał, Musial, Kazimierz
Economics, Econometrics and Finance · Mathematics · #46G10 #54C60 #Advanced Banach Space Theory #FOS: Mathematics #Functional Analysis (math.FA) #Housing Market and Economics #Primary 28B20 Secondary 28B05
paper · pdf · doi:10.48550/arxiv.2305.19653
openalex publication_date 2023/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be a Banach space with RNP, (\vO,\vS,μ) be a complete probability space and \vG:\vO→cb(X) (nonempty, closed convex and bounded subsets of X) be a multifunction. Assume that \vX⊂\vS is a σ-algebra and the multimeasure M defined by the Pettis integral of \vG be such that the restriction of M to \vX is of σ-finite variation. Using a lifting, I prove the existence of an Effros measurable conditional expectation of \vG and present its representation in terms of quasi-selections of \vG. I apply then the description to martingales of Pettis integrable multifunctions obtaining a scalarly equivalent martingale of measurable multifunctions with many martingale selections. In general the situation cannot be reduced to the separable space.