Previous |  Up |  Next


asymptotic density; measure; ultrafilter; P-ultrafilter
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., {A Note on extensions of asymptotic density}, Proc. Amer. Math. Soc. {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.
[1] Bartoszyński T., Judah H.: Set Theory: On the Structure of the Real Line. A. K. Peters, Wellesley, 1995. MR 1350295
[2] Blass A., Frankiewicz R., Plebanek G., Ryll-Nardzewski C.: A Note on extensions of asymptotic density. Proc. Amer. Math. Soc. 129 (2001), no. 11, 3313–3320. DOI 10.1090/S0002-9939-01-05941-X | MR 1845008
[3] Fremlin D. H.: Measure Theory, Vol. 3: Measure Algebras. Torres Fremlin, Colchester, 2004. MR 2459668
[4] Kunisada R.: Density measures and additive property. J. Number Theory 176 (2017), 184–203. DOI 10.1016/j.jnt.2016.12.013 | MR 3622126
[5] Smith E. C. Jr., Tarski A.: Higher degrees of distributivity and completeness in Boolean algebras. Trans. Amer. Math. Soc. 84 (1957), 230–257. DOI 10.1090/S0002-9947-1957-0084466-4 | MR 0084466
Partner of
EuDML logo