Absolute continuity, singularity and product measures

Roy A Johnson

doi:10.1155/s0161171282000738

Abstract

Conditions are given under which a product of two semifinite measures is absolutely continuous or weakly singular with respect to another product of two semifinite measures. A Lebesgue type decomposition theorem is proved for certain product measures so that the resulting measures are themselves product measures.

Highlights

The main results of this paper are Theorems 4.2 and 4.3 in which we observe conditions under which the Lebesgue decomposition of one product measure with respect to another yields measures which are themselves product measures
It will be seen that the expected results hold for the smallest product of two semifinite measures but that some surprising things can happen for the largest product of two semifinite measures
In view of the Lebesgue Decomposition Theorem, we shall see that a product measure ( x 9) can be written as the sum of two measures so that the frst is absolutely continuous with respect to a product measure v)##. (H’ V) ## and the second is weakly singular with respect to

Summary

INTRODUCTION

The main results of this paper are Theorems 4.2 and 4.3 in which we observe conditions under which the Lebesgue decomposition of one product measure with respect to another yields measures which are themselves product measures. ( x it is clear that is itself a product of and i 2 THEOREM i.I. Suppose and are measures on g and 9 is a measure on g. To prove (i) notice that the four measures agree on measurable rectangles F x G such that (I(F) + 2(F))9(G) is finite. )# 9,)L 9)#.’ [resp., 9] is semifinite (and o-finite) o.n any set for which. O. By hypothesis, and 9 are semifinite and o-finite on G and H, respectively. 9 are semifinite and o-finite on G and H, respectively S Let denote the smallest (semifinite) measure agreeing with on the sets for which is finite Corollary 2.4 is completely expected, its proof is surprisingly complicated without the use of Theorem 2.3

WEAK SINGULARITY AND PRODUCT MEASURES

PRODUCT MEASURES AND THE LEBESGUE DECOMPOSITION

Findings

COMPLEMENTS ON STRONG-RECESSIVENESS AND QUASI-DOMINANCE