TY - JOUR

T1 - (Non-)Distributivity of the Product for σ-Algebras with Respect to the Intersection

AU - Steinicke, Alexander

PY - 2021/2/22

Y1 - 2021/2/22

N2 - We study the validity of the distributivity equation (A⊗F)∩(A⊗G)=A⊗(F∩G),where A is a σ-algebra on a set X, and F, G are σ-algebras on a set U. We present a counterexample for the general case and in the case of countably generated subspaces of analytic measurable spaces, we give an equivalent condition in terms of the σ-algebras’ atoms. Using this, we give a sufficient condition under which distributivity holds.

AB - We study the validity of the distributivity equation (A⊗F)∩(A⊗G)=A⊗(F∩G),where A is a σ-algebra on a set X, and F, G are σ-algebras on a set U. We present a counterexample for the general case and in the case of countably generated subspaces of analytic measurable spaces, we give an equivalent condition in terms of the σ-algebras’ atoms. Using this, we give a sufficient condition under which distributivity holds.

UR - http://www.scopus.com/inward/record.url?scp=85101239773&partnerID=8YFLogxK

U2 - 10.1007/s00013-020-01571-z

DO - 10.1007/s00013-020-01571-z

M3 - Article

SN - 0003-889X

JO - Archiv der Mathematik

JF - Archiv der Mathematik

ER -