Abstract
Given a measurable space (T, F), a set X, and a map ϕ: T → X, the σ-algebras 
N
Ф = ⋂ϕ∈Φ
N
ϕ, and M
Φ = ⋂ϕ∈Φ
N
ϕ, where G
ϕ(t) = (t, ϕ(t)) and Φ ⊂ X
T, are considered. These σ-algebras are used to characterize the (F, B, ℬ)-measurability of the compositions g ○ ϕ and f о G
ϕ, where g: X → Y, f: T × X → Y, and (Y, ℬ) is a measurable space. Their elements are described without using the operations ϕ
−1 and G
−1ϕ
.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Bibliography
I. V. Shragin, “Measurability conditions for superpositions,” Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.], 197 (1971), no. 2, 295–298.
I. V. Shragin, “Superposition measurability,” Izv. Vyssh. Uchebn. Zaved. Mat. [Soviet Math. (Iz. VUZ)] (1975), no. 1, 82–92.
J. Appell and P. P. Zabrejko, Nonlinear Superposition Operators, Cambridge Univ. Press, Cambridge, 1990.
Yu. V. Nepomnyashchikh and A. V. Ponosov, “Local operators in some subspaces of the space L 0,” Izv. Vyssh. Uchebn. Zaved. Mat. [Russian Math. (Iz. VUZ)] (1999), no. 6, 50–64.
Additional information
__________
Translated from Matematicheskie Zametki, vol. 80, no. 6, 2006, pp. 926–933.
Original Russian Text Copyright © 2006 by I. V. Shragin.
Rights and permissions
About this article
Cite this article
Shragin, I.V. On σ-algebras related to the measurability of compositions. Math Notes 80, 868–874 (2006). https://doi.org/10.1007/s11006-006-0209-1
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11006-006-0209-1