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- W2794519177 abstract "A family $$mathcal {F}subseteq left[ omega right] ^omega $$ is called Rosenthal if for every Boolean algebra $$mathcal {A}$$ , bounded sequence $$big langle mu _k: kin omega big rangle $$ of measures on $$mathcal {A}$$ , antichain $$big langle a_n: nin omega big rangle $$ in $$mathcal {A}$$ , and $$varepsilon >0$$ , there exists $$Ain mathcal {F}$$ such that $$sum _{nin A, nne k}mu _k(a_n)<varepsilon $$ for every $$kin A$$ . Well-known and important Rosenthal’s lemma states that $$left[ omega right] ^omega $$ is a Rosenthal family. In this paper we provide a necessary condition in terms of antichains in $${wp (omega )}$$ for a family to be Rosenthal which leads us to a conclusion that no Rosenthal family has cardinality strictly less than $${{mathrm{cov}}}(mathcal {M})$$ , the covering of category. We also study ultrafilters on $$omega $$ which are Rosenthal families—we show that the class of Rosenthal ultrafilters contains all selective ultrafilters (and consistently selective ultrafilters comprise a proper subclass)." @default.
- W2794519177 created "2018-04-06" @default.
- W2794519177 creator A5084059925 @default.
- W2794519177 date "2018-03-27" @default.
- W2794519177 modified "2023-09-24" @default.
- W2794519177 title "Families of sets related to Rosenthal’s lemma" @default.
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- W2794519177 doi "https://doi.org/10.1007/s00153-018-0621-8" @default.
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