A class of $P^t(d\mu )$ spaces whose point evaluations vary with $t$

Extending an example given by T. Kriete, we develop a class of measures each of which consists of a measure on $\{z:\,|z|{}=1\}$ along with a series of weighted point masses in $\mathbf{D:=}\{z:\,|z|{}<1\}$. This class provides relatively simple examples of measures $\mu$ which have the property that the collection of analytic bounded point evaluations for $P^{t}(d\mu )$ varies with $t$. The first known measures with this property were recently constructed by J. Thomson.