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 -