Generalized Densities on ℝ n and their Applications

We examine some generalized densities (called (ψ, n)-densities) obtained as a result of strengthening the Lebesgue Density Theorem. It turns out that these notions are the generalizations of superdensity, enhanced density and m-density, and have some applications in the theory of sets of finite perimeter and in Sobolev spaces.

On Generalization of the 𝒯AI-Density Topology

We present a further generalization of the 𝒯AI-density topology introduced in [W. Wojdowski, A category analogue of the generalization of Lebesgue density topology, Tatra Mt. Math. Publ. 42 (2009), 11–25] as a generalization of the I-density topology. We construct an ascending sequence {𝒯AI(n)}n∈ℕ of density topologies which leads to the 𝒯AI(ω)-density topology including all previous topologies. We examine several basic properties of the topologies.

Corrigendum to a Category Analogue of The Generalization of Lebesgue Density Topology

The notion of AI -density point introduced in Wojdowski, W. A topology stronger than the Lebesgue density topology, in: Real Functions, Density Topology and Related Topics, Łódź Univ. Press, 2011, pp. 73-80 [WO1]. leads to the operator ΦAI (A) which is not a lower density operator. We present a counterexample giving a corrected definition which should be used in [WO1] to keep all results valid.

LEBESGUE DENSITY AND CLASSES

Analyzing the effective content of the Lebesgue density theorem played a crucial role in some recent developments in algorithmic randomness, namely, the solutions of the ML-covering and ML-cupping problems. Two new classes of reals emerged from this inquiry: thepositive density pointswith respect toeffectively closed(or$\prod _1^0$) sets of reals, and a proper subclass, thedensity-one points. Bienvenu, Hölzl, Miller, and Nies have shown that the Martin-Löf random positive density points are exactly the ones that do not compute the halting problem. Treating this theorem as our starting point, we present several new results that shed light on how density, randomness, and computational strength interact.

Explicit Error Bounds via Total Variation

This chapter concerns the obtaining of explicit error estimates for convergence to Benford's law, with an analysis done through the total variation of the densities. This method yields reasonable estimates for Benford's law in many cases, and is often simpler to calculate and more elementary than Fourier methods. Here, the chapter provides the distribution of the remainder U in the case of Y having a Lebesgue density f, defines the measures of non-uniformity of this distribution, and collects some basic facts about the total variation of functions. The main results, examples, and proofs are then presented in the final three sections of this chapter.

On the generalization of density topologies on the real line

The paper concerns the density points with respect to the sequences of intervals tending to zero in the family of Lebesgue measurable sets. It shows that for some sequences analogue of the Lebesgue density theorem holds. Simultaneously, this paper presents proof of theorem that for any sequence of intervals tending to zero a relevant operator ϕJ generates a topology. It is almost but not exactly the same result as in the category aspect presented in [WIERTELAK, R.: A generalization of density topology with respect to category, Real Anal. Exchange 32 (2006/2007), 273–286]. Therefore this paper is a continuation of the previous research concerning similarities and differences between measure and category.

Generalized Discontinuity of Real-Valued Functions

We present a proof of the theorem on countability of the set of points of generalized discontinuity of an ( , ) regular real function f:X→ℝ, where is a local system in X and is a partition of X. We start with a definition of a local system in a generalized form and with basic properties of local systems. The concepts are illustrated with examples. The main result is applied both for regularities in the sense of density connected with the Lebesgue measure on ℝn (Lebesgue density) and with Baire category -density), respectively.