## Jan GrebíkUltrafilter extensions of asymptotic density

Comment.Math.Univ.Carolin. 60,1 (2019) 25-37.

Abstract:We characterize for which ultrafilters on $\omega$ is the ultrafilter extension of the asymptotic density on natural numbers $\sigma$-additive on the quotient boolean algebra $\mathcal{P}(\omega)/d_{\mathcal{U}}$ or satisfies similar additive condition on $\mathcal{P}(\omega)/\text{fin}$. These notions were defined in [Blass A., Frankiewicz R., Plebanek G., Ryll-Nardzewski~C., {\it A~Note on extensions of asymptotic density}, Proc.\ Amer.\ Math.\ Soc.\ {\bf 129} (2001), no.\ 11, 3313--3320] under the name ${\boldsymbol{AP}}$(null) and ${\boldsymbol{AP}}$(*). We also present a~characterization of a $P$- and semiselective ultrafilters using the ultraproduct of $\sigma$-additive measures.

Keywords: asymptotic density; measure; ultrafilter; P-ultrafilter

DOI: DOI 10.14712/1213-7243.2015.279
AMS Subject Classification: 28A12 03E05 03E35 11B05

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