# Article

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