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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 $𝒫\left(\omega \right)/d_{𝒰}$ or satisfies similar additive condition on $𝒫\left(\omega \right)/\text{fin}$. 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 $AP$(null) and $AP$(*). We also present a characterization of a $P$- and semiselective ultrafilters...