| Title:
|
Ultrafilter extensions of asymptotic density (English) |
| Author:
|
Grebík, Jan |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
60 |
| Issue:
|
1 |
| Year:
|
2019 |
| Pages:
|
25-37 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
asymptotic density |
| Keyword:
|
measure |
| Keyword:
|
ultrafilter |
| Keyword:
|
P-ultrafilter |
| MSC:
|
03E05 |
| MSC:
|
03E35 |
| MSC:
|
11B05 |
| MSC:
|
28A12 |
| idZBL:
|
Zbl 07088823 |
| idMR:
|
MR3946662 |
| DOI:
|
10.14712/1213-7243.2015.279 |
| . |
| Date available:
|
2019-05-13T07:43:53Z |
| Last updated:
|
2021-04-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147672 |
| . |
| Reference:
|
[1] Bartoszyński T., Judah H.: Set Theory: On the Structure of the Real Line.A. K. Peters, Wellesley, 1995. MR 1350295 |
| Reference:
|
[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. MR 1845008, 10.1090/S0002-9939-01-05941-X |
| Reference:
|
[3] Fremlin D. H.: Measure Theory, Vol. 3: Measure Algebras.Torres Fremlin, Colchester, 2004. MR 2459668 |
| Reference:
|
[4] Kunisada R.: Density measures and additive property.J. Number Theory 176 (2017), 184–203. MR 3622126, 10.1016/j.jnt.2016.12.013 |
| Reference:
|
[5] Smith E. C. Jr., Tarski A.: Higher degrees of distributivity and completeness in Boolean algebras.Trans. Amer. Math. Soc. 84 (1957), 230–257. MR 0084466, 10.1090/S0002-9947-1957-0084466-4 |
| . |
