A continuity property of the convolution (Q1095483)

Let \({\mathcal B}\) be the Borel \(\sigma\)-field of R. The author proves the following theorem: If \(P_ n\), \(n\geq 1\), is a sequence of probability measures on \({\mathcal B}\) converging weakly to \(P_ 0\), then \(P_ n*P\to^{\omega^ 2}P_ 0*P\) for all probability measures P on \({\mathcal B}\), where \(\to^{\omega^ 2}\) means convergence in the sense of \textit{J. K. Brooks} and \textit{R. V. Chacon} [Adv. Math. 37, 16-26 (1980; Zbl 0463.28003)]. Corollary: Let P be a probability measure on \({\mathcal B}\). Then (i) for any sequence \(x_ n\in R\), \(n\geq 1\), satisfying \(x_ n\to x\in R\), \[ P*\delta_{x_ n}(B)\to P*\delta_ x(B)\text{ for all } B\in {\mathcal B} \] iff \(P*\delta_{x_ n}\), \(n\geq 1\), are uniformly absolutely continuous with respect to a \(\sigma\)-finite measure on \({\mathcal B}\), and (ii) \(P*\delta_{x_ n}(B)\to P*\delta_ x(B)\) for all \(B\in {\mathcal B}\) and any sequence \(x_ n\in R\), \(n\geq 1\), satisfying \(x_ n\to x\in R\) iff \(P*\delta_ x\), \(x\in R\), are uniformly absolutely continuous with respect to the Lebesgue measure.