We show that, whenever $\Gamma$ is a countable abelian group and $\Delta$ is a finitely-generated subgroup of $\Gamma$, a generic measure-preserving action of $\Delta$ on a standard atomless probability space $(X,\mu )$ extends to a free measure-preserving action of $\Gamma$ on $(X,\mu )$. This extends a result of Ageev, corresponding to the case when $\Delta$ is infinite cyclic.

Players ONE and TWO play the following game: In the nth inning ONE chooses a set $O_n$ from a prescribed family ℱ of subsets of a space X; TWO responds by choosing an open subset $T_n$ of X. The players must obey the rule that $O_n\subseteq O_{n+1}\subseteq T_{n+1}\subseteq T_n$ for each n. TWO wins if the intersection of TWO’s sets is equal to the union of ONE’s sets. If ONE has no winning strategy, then each element of ℱ is a $G_{\delta }$-set. To what extent is the converse true? We show that:  (A) For ℱ the collection of countable subsets of X:   1. There are subsets...

Based on some earlier findings on Banach Category Theorem for some “nice” $\sigma$-ideals by J. Kaniewski, D. Rose and myself I introduce the $h$ operator ($h$ stands for “heavy points”) to refine and generalize kernel constructions of A. H. Stone. Having obtained in this way a generalized Kuratowski’s decomposition theorem I prove some characterizations of the domains of functions having “many” points of $h$-continuity. Results of this type lead, in the case of the $\sigma$-ideal of meager sets, to important statements...

R. A. Johnson showed that there is no translation-invariant Borel lifting for the measure algebra of ℝ/ℤ equipped with Haar measure, a result which was generalized by M. Talagrand to non-discrete locally compact abelian groups and by J. Kupka and K. Prikry to arbitrary non-discrete locally compact groups. In this paper we study analogs of these results for category algebras (the Borel σ-algebra modulo the ideal of first category sets) of topological groups. Our main results are for the class of...

ℒ denotes the Lebesgue measurable subsets of ℝ and ${ℒ}_0$ denotes the sets of Lebesgue measure 0. In 1914 Burstin showed that a set M ⊆ ℝ belongs to ℒ if and only if every perfect P ∈ ℒ$ℒ0$hasaperfectsubsetQ\in ℒ\${ℒ}_0$ which is a subset of or misses M (a similar statement omitting “is a subset of or” characterizes ${ℒ}_0$). In 1935, Marczewski used similar language to define the σ-algebra (s) which we now call the “Marczewski measurable sets” and the σ-ideal $\left(s^0\right)$ which we call the “Marczewski null sets”. M ∈ (s) if every perfect set P has...

We prove that, assuming MA, every crowded $T_0$ space $X$ is $\omega$-resolvable if it satisfies one of the following properties: (1) it contains a $\pi$-network of cardinality $<𝔠$ constituted by infinite sets, (2) $\chi \left(X\right)<𝔠$, (3) $X$ is a $T_2$ Baire space and $c\left(X\right)\le {\aleph }_0$ and (4) $X$ is a $T_1$ Baire space and has a network $𝒩$ with cardinality $<𝔠$ and such that the collection of the finite elements in it constitutes a $\sigma$-locally finite family. Furthermore, we prove that the existence of a $T_1$ Baire irresolvable space is equivalent to the existence of...

We show that the existence of a non-trivial category base on a set of regular cardinality with each subset being Baire is equiconsistent to the existence of a measurable cardinal.

Let C denote the Banach space of real-valued continuous functions on [0,1]. Let Φ: C × C → C. If Φ ∈ +, min, max then Φ is an open mapping but the multiplication Φ = · is not open. For an open ball B(f,r) in C let B²(f,r) = B(f,r)·B(f,r). Then f² ∈ Int B²(f,r) for all r > 0 if and only if either f ≥ 0 on [0,1] or f ≤ 0 on [0,1]. Another result states that Int(B₁·B₂) ≠ ∅ for any two balls B₁ and B₂ in C. We also prove that if Φ ∈ +,·,min,max, then the set ${\Phi }^{-1}\left(E\right)$ is residual whenever E is residual in...

Let be the class of Banach spaces X for which every weakly quasi-continuous mapping f: A → X defined on an α-favorable space A is norm continuous at the points of a dense $G_{\delta }$ subset of A. We will show that this class is stable under c₀-sums and ${\ell }^p$-sums of Banach spaces for 1 ≤ p < ∞.