We show that for a wide class of σ-algebras 𝓐, indicatrices of 𝓐-measurable functions admit the same characterization as indicatrices of Lebesgue-measurable functions. In particular, this applies to functions measurable in the sense of Marczewski.

We show that the classes of separable reflexive Banach spaces and of spaces with separable dual are strongly bounded. This gives a new proof of a recent result of E. Odell and Th. Schlumprecht, asserting that there exists a separable reflexive Banach space containing isomorphic copies of every separable uniformly convex Banach space.

We prove the following theorems: There exists an $\omega$-covering with the property $s_0$. Under $\mathrm{c}ov\left(𝒩\right)=$ there exists $X$ such that ${\forall }_{B\in ℬor}[B\cap X$ is not an $\omega$-covering or $X\setminus B$ is not an $\omega$-covering]. Also we characterize the property of being an $\omega$-covering.

We give a different proof of the well-known fact that any uncountable family of analytic subsets of a Polish space with the point-finite intersection property must contain a subfamily whose union is not analytic. Our approach is based on the Kunen-Martin theorem.

We show that ω₁-Universally Baire self-justifying systems are fully Universally Baire under the Weak Stationary Reflection Principle for Pairs. This involves analyzing the notion of a weakly captured set of reals, a weakening of the Universal Baire Property.

Let (FP) abbreviate the statement that ${\int }_0^1\left({\int }_0^{1}fdy\right)dx={\int }_0^1\left({\int }_0^{1}fdx\right)dy$ holds for every bounded function f: [0,1]² → ℝ whenever each of the integrals involved exists. We shall denote by (SFP) the statement that the equality above holds for every bounded function f: [0,1]² → ℝ having measurable vertical and horizontal sections. It follows from well-known results that both of (FP) and (SFP) are independent of the axioms of ZFC. We investigate the logical connections of these statements with several other strong Fubini type properties...

 Let I and J be σ-ideals on Polish spaces X and Y, respectively. We say that the pair ⟨I,J⟩ has the Strong Fubini Property (SFP) if for every set D ⊆ X× Y with measurable sections, if all its sections $D_x=y:⟨x,y⟩\in D$ are in J, then the sections $D^y=x:⟨x,y⟩\in D$ are in I for every y outside a set from J (“measurable" means being a member of the σ-algebra of Borel sets modulo sets from the respective σ-ideal). We study the question of which pairs of σ-ideals have the Strong Fubini Property. Since CH excludes this phenomenon completely,...

We analyze several “strong meager” properties for filters on the natural numbers between the classical Baire property and a filter being $F_{\sigma }$. Two such properties have been studied by Talagrand and a few more combinatorial ones are investigated. In particular, we define the notion of a P⁺-filter, a generalization of the traditional concept of P-filter, and prove the existence of a non-meager P⁺-filter. Our motivation lies in understanding the structure of filters generated by complements of members...

Let $X\subseteq 2^w$. Consider the class of all Borel $F\subseteq X\times 2^w$ with null vertical sections $F_x$, x ∈ X. We show that if for all such F and all null Z ⊆ X, ${\cup }_{x\in Z}{F}_x$ is null, then for all such F, ${\cup }_{x\in X}{F}_x\ne 2^w$. The theorem generalizes the fact that every Sierpiński set is strongly meager and was announced in [P].

Nous donnons, pour une certaine catégorie de boréliens d'un produit de deux espaces polonais, comprenant les boréliens à coupes dénombrables, une caractérisation du type "test d'Hurewicz" de ceux ne pouvant pas être rendus différence transfinie d'ouverts par changement des deux topologies polonaises.

The set of squares in the group of autohomeomorphisms of the circle is complete analytic, and hence analytic but not Borel.

The covering property for σ-ideals of compact sets is an abstract version of the classical perfect set theorem for analytic sets. We will study its consequences using as a paradigm the σ-ideal of countable closed subsets of $2^{\omega }$.

Let K be a subclass of Mod() which is closed under isomorphism. Vaught showed that K is ${\Sigma }_{\alpha }$ (respectively, ${\Pi }_{\alpha }$) in the Borel hierarchy iff K is axiomatized by an infinitary ${\Sigma }_{\alpha }$ (respectively, ${\Pi }_{\alpha }$) sentence. We prove a generalization of Vaught’s theorem for the effective Borel hierarchy, i.e. the Borel sets formed by union and complementation over c.e. sets. This result says that we can axiomatize an effective ${\Sigma }_{\alpha }$ or effective ${\Pi }_{\alpha }$ Borel set with a computable infinitary sentence of the same complexity. This result...

It is known that there is a comeagre set of mutually conjugate measure preserving homeomorphisms of Cantor space equipped with the coinflipping probability measure, i.e., Haar measure. We show that the generic measure preserving homeomorphism is moreover conjugate to all of its powers. It follows that the generic measure preserving homeomorphism extends to an action of (ℚ, +) by measure preserving homeomorphisms, and, in fact, to an action of the locally compact ring 𝔄 of finite adèles. ...