Realizing automorphisms of quotients of product σ-fields

Siegfried Graf

Vol. 99 (1982), No. 1, 19–30

DOI: 10.2140/pjm.1982.99.19

Abstract

Let (X_α)_α∈I be a family of Polish spaces, X = Π_α∈IX_α, and B the product of the Borel fields of the spaces X_α. For K ⊂ I let X_K = Π_α∈KX_α and let π_K : X → X_K be the canonical projection. Moreover, let n be a σ-ideal in B satisfying the following Fubini type condition:

N ∈ n if and only if π_J⁻¹({z ∈ X_J|π_I∖J⁻¹({y ∈ X_I∖J|(z,y) ∈ N})∉n}) ∈ n for every nonempty J ⊂ I. Then, given an automorphism Φ from B∕n onto itself, there exists a bijection f : X → X such that f and f⁻¹ are measurable and

for all B ∈ B.

Mathematical Subject Classification 2000

Primary: 28A35

Secondary: 54H13

Milestones

Received: 28 March 1980

Published: 1 March 1982