(Non-)Distributivity of the Product for σ-Algebras with Respect to the Intersection
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In: Archiv der Mathematik, Vol. 116, No. 6, 06.2021, p. 667-675.
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TY - JOUR
T1 - (Non-)Distributivity of the Product for σ-Algebras with Respect to the Intersection
AU - Steinicke, Alexander
N1 - Publisher Copyright: © 2021, The Author(s).
PY - 2021/6
Y1 - 2021/6
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
VL - 116
SP - 667
EP - 675
JO - Archiv der Mathematik
JF - Archiv der Mathematik
SN - 0003-889X
IS - 6
ER -
