Introduction. In [ 11 BROWN obtained necessary and sufficient conditions for the singularity of POISSON processes P, and P,, with a-finite mean measures v and p. In this paper we show for RADON measures Y and p that P, and P,, are singular iff P,/%and P,,/%,., are singular. Particularly we find a set A E % .
, with P J A ) = 1 and P,(A) = 0.
- Let X be a fixed locally compact second countable HATJSDORFF-space, Z be the BOREL a-algebra in X and 8 be the set of all bounded sets in Z. Let N be the set of all integer valued RADON measures on (X, Z) and % be the a-algebra in N generated by the mappings y-y(B) (BE 8). For each VE 8 we denote by !Rev the a-algebra in N generated by the mappings y*y(B) (BE 8, BC CV) and by %-: = n %cv. Furthermore, let Y and p be RADON measures on (X, Z), let p = A + o be the LEBESQUE decomposition of p with respect to v (A< c v , o I v ) with f : =-and let U E Z be a set with Y ( U) = 0 and w(CU) = 0. let P , and P , be POISSON processes with mean measurea v and p. Then P , I P, iff one of the following three conditions hold: V'EB d l dv L In [l] it is shown: A: w ( X ) == B : j If-11 dv=m for some C > O ( 1 1 -1 I c) C : J (f-1)2dY=m forall c -0 .
{ I f -l l S C )
Now in each case we shall find a set A E 8with P,,(A) = 1 and P J A ) = 0.
Let V i z V2s. . . C 8 be a, fixed sequence with nEN 0) u V7,=X, (ii) For each BE 8 there exibts a nE N such that B S V,, .