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.
In this paper we define certain types of projections of planar sets and study some properties of such projections.
We prove the following theorems: There exists an -covering with the property . Under there exists such that is not an -covering or is not an -covering]. Also we characterize the property of being an -covering.
It is shown that if A ⊂ ℝ has the same constant shade with respect to all Banach measures, then the same is true of any similarity transformation of A and the shade is not changed by the transformation. On the other hand, if A ⊂ ℝ has constant μ-shade with respect to some fixed Banach measure μ, then the same need not be true of a similarity transformation of A with respect to μ. But even if it is, the μ-shade might be changed by the transformation. To prove such a μ exists, a Hamel basis with some...
Let (FP) abbreviate the statement that 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 are in J, then the sections 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,...