Berger measure for S(a,b,c,d)

In this note, we get the Berger measure of the weighted shift S(a,b,c,d) with weights αn:=an+bcn+d (a,b,c,d>0 and n⩾0) as well as its p-subshift. Then we will give examples from analytic function theory to illustrate that the necessary condition given by Curto and Yoon (2006) [12] for subnormality of 2-variable weighted shift is not sufficient. Furthermore, we will do some calculations about subnormal backward extension of S(a,b,c,d).