Two Examples Related to Properties of Discrete Measures

Two examples illustrating properties of discrete measures are given. In the first part of the paper, it is proved that, for any probability measure $\mu$ with $\operatorname{supp}{\mu}=[-1,1]$ whose logarithmic potential is continuous on $[-1,1]$, there exists a (discrete) measure $\sigma=\sigma(\mu)$ with $\operatorname{supp}{\sigma}=[-1,1]$ such that the corresponding orthogonal polynomials $P_n(x;\sigma)=x^n+\dotsb$ satisfy the condition $(1/n)\chi(P_n(\,\cdot\,;\sigma))\xrightarrow{*}\mu$, $n\to\infty$, where $\chi(\,\cdot\,)$ is the measure counting the zeros of a polynomial. The proof of the existence of such a measure $\sigma$ is based on properties of weighted Leja points. In the second part, an example of a compact set and a sequence of discrete measures supported on it with a special property is given. Namely, the sequence of measures converges in the $*$-weak topology to the equilibrium measure on the compact set, but the corresponding sequence of logarithmic potentials converges in capacity to the equilibrium potential in no neighborhood of this compact set.