Full Outer Measure Research Articles

Families of feebly continuous functions and their properties

Let f:R2→R. The notions of feeble continuity and very feeble continuity of f at a point 〈x,y〉∈R2 were considered by I. Leader in 2009. We study properties of the sets FC(f) (respectively, VFC(f)⊃FC(f)) of points at which f is feebly continuous (very feebly continuous). We prove that VFC(f) is densely nonmeager, and, if f has the Baire property (is measurable), then FC(f) is residual (has full outer Lebesgue measure). We describe several examples of functions f for which FC(f)≠VFC(f). Then we consider the notion of two-feeble continuity which is strictly weaker than very feeble continuity. We prove that the set of points where (an arbitrary) f is two-feebly continuous forms a residual set of full outer measure. Finally, we study the existence of large algebraic structures inside or outside various sets of feebly continuous functions.

A transversal of full outer measure

We show that for every partition of a set of reals into countable sets there is a transversal of the same outer measure.

NONMEASURABLE SETS WITH REGULAR SECTIONS

We improve and generalize the result of Kirchheim and Natkaniec \cite{kirchheim} by proving that there exists $D \subset [0,\pi)$ of full outer measure such that the class $\bigcap_{\theta \in D}\mathcal{G}_2(\theta) \cap \bigcap_{\theta \in [0,\pi)}\mathcal{G}_3(\theta)$ contains a nonmeasurable set. The present result is obtained within ZFC.

Generalized Egoroff’s theorem

Abstract This note is closely related to the paper [R. Pinciroli: On theindependence of a generalized statement of Egoroff’s theorem from ZFC afterT. Weiss, Real Anal. Exchange 32 (2006-2007), 225-232] and it presents slight improvements of its results. Theorem 1.13 shows a connection with Galois-Tukey embeddings; Corollary 1.14 presents another inequality which is dual to the previously known one; Corollary 3.5 shows that there is no distinction between positive outer measure and full outer measure in the given context; and Corollary 4.3 unifies the known counterexamples.

On the Independence of a Generalized Statement of Egoroff's Theorem from ZFC after T. Weiss

We consider a generalized version (GES) of the wellknown Severini-Egoroff theorem in real analysis, first shown to be undecidable in ZFC by Tomasz Weiss. This independence is easily derived from suitable hypotheses on some cardinal characteristics of the continuum like b and o, the latter being the least cardinality of a subset of [0,1] having full outer measure.

SUPPORT OF A JOINT RESOLUTION OF IDENTITY AND THE PROJECTION SPECTRAL THEOREM

Let A = (Ax)x ∈ Xbe a family of commuting normal operators in a separable Hilbert space H0. Obtaining the spectral expansion of A involves constructing of the corresponding joint resolution of identity E. The support supp E is not, in general, a set of full measure. This causes numerous difficulties, in particular, when proving the projection spectral theorem, i.e. the main theorem about the expansion in generalized joint eigenvectors. In this work, we show that supp E has a full outer measure under the conditions of the projection spectral theorem. Using this result, we simplify the proof of the theorem and refine its assertions.

Solutions with big graph of an equation of the second iteration

We consider the functional equation¶\( \varphi(f(\varphi(x)))=g(x,\varphi(x)) \)¶and we look for its solutions which have a big graph. The graph of such a solution has some strange properties: it has full outer measure and is topologically big.

Symmetric continuity of real functions

It is shown that the set of points where a real function is both symmetrically continuous and not continuous has inner measure zero but may have full outer measure.

Symmetric Continuity of Real Functions

It is shown that the set of points where a real function is both symmetrically continuous and not continuous has inner measure zero but may have full outer measure.