所有普遍可测集都可以是Δ^1_2集

Universally measurable subgroups of countable index

It is proved that any countable index, universally measurable subgroup of a Polish group is open. By consequence, any universally measurable homomorphism from a Polish group into the infinite symmetric group $S_\infty$ is continuous. It is also shown that a universally measurable homomorphism from a Polish group into a second countable, locally compact group is necessarily continuous.

Universally measurable sets in generic extensions

A subset of a topological space is said to be \emph{universally measurable} if it is measured by the completion of each countably additive $σ$-finite Borel measure on the space, and \emph{universally null} if it has measure zero for each such atomless measure. In 1908, Hausdorff proved that there exist universally null sets of real numbers of cardinality $\aleph_{1}$, and thus that there exist at least $2^{\aleph_{1}}$ such sets. Laver showed in the 1970's that consistently there are just continuum many universally null sets of reals. The question of whether there exist more than continuum man…

Measurable cardinals and the cardinality of Lindelöf spaces

If it is consistent that there is a measurable cardinal, then it is consistent that all points g-delta Rothberger spaces have "small" cardinality.

一个学习问题，其一致性等价于实值可测基数的不存在性

A learning problem whose consistency is equivalent to the non-existence of real-valued measurable cardinals

本文证明 k-最近邻学习规则的普遍一致性与实值可测基数的不存在性等价，揭示了机器学习一致性问题与集合论大基数假设之间的深刻联系。

The Filter Dichotomy and medial limits

The \emph{Filter Dichotomy} says that every uniform nonmeager filter on the integers is mapped by a finite-to-one function to an ultrafilter. The consistency of this principle was proved by Blass and Laflamme. A function between topological spaces is \emph{universally measurable} if the preimage of %every open subset of the codomain is measured by every Borel measure on the domain. A \emph{medial limit} is a universally measurable function from $\mathcal{P}(ω)$ to the unit interval [0,1] which is finitely additive for disjoint sets, and maps singletons to 0and $ω$ to 1. Christensen and Mokobod…

Naively Haar null sets in Polish groups

Let $(G,\cdot)$ be a Polish group. We say that a set $X \subset G$ is Haar null if there exists a universally measurable set $U \supset X$ and a Borel probability measure $μ$ such that for every $g, h \in G$ we have $μ(gUh)=0$. We call a set $X$ naively Haar null if there exists a Borel probability measure $μ$ such that for every $g, h \in G$ we have $μ(gXh)=0$. Generalizing a result of Elekes and Steprāns, which answers the first part of Problem FC from Fremlin's list, we prove that in every abelian Polish group there exists a naively Haar null set that is not Haar null.