This paper provides a list of statements of single-variable Real Analysis, including well-known theorems, that are equivalent to the completeness or Archimedean properties of totally ordered fields. There are 72 characterizations of completeness and 42 characterizations of the Archimedean property, among them many that appear to be new in the sense that they do not seem to have previously been mentioned in this context. An attempt is made to be as comprehensive as possible and to give a complete account of the current state of knowledge of the matter. Proofs are provided whenever they are not readily available in the literature.

Local oscillation of a function satisfying a Hölder condition is considered, and it is proved that its growth is governed by a version of the Law of the Iterated Logarithm.

The space \(F(\ell_2)\) of all closed subsets of \(\ell_2\) is a Polish space. We show that the subset \(P\subset F(\ell_2)\) consisting of the purely \(1\)-unrectifiable sets is \(\Pi_1^1\)-complete.

Let \(X\) be a Baire space, \(Y\) a topological space, \(Z\) a regular space and \(f:X \times Y \to Z\) be a horizontally quasi-continuous function. We will show that if \(Y\) is first countable and \(f\) is quasi-continuous with respect to the first variable, then every horizontally quasi-continuous function \(f:X \times Y \to Z\) is jointly quasi-continuous. This will extend Martin’s Theorem of quasi-continuity of separately quasi-continuous functions for non-metrizable range. Moreover, we will prove quasi-continuity of \(f\) for the case \(Y\) is not necessarily first countable.

Let \(\Omega\) be a polygonal domain in \(\mathbb{R}^2\) and let \(U\) be a weak solution of \( -\Delta u=f\) in \( \Omega\) with Dirichlet boundary condition, where \(f\in L^p_\omega(\Omega)\) and \(\omega\) is a weight in \(A_p(\mathbb{R}^2)\), \(1<p<\infty\). We give some estimates of the Green function associated to this problem involving some functions of the distance to the vertices and the angles of \(\Omega\). As a consequence, we can prove an a priori estimate for the solution \(u\) on the weighted Sobolev spaces \(W^{2,p}_\omega(\Omega)\), \(1<p<\infty\).

Let \[ S_p(a,b;t)=\frac{1}{b}\sum_{k=0}^{p} \frac{{p\choose k}}{ {ak+b \choose b} } t^k, \] with \(p\in \mathbb{N}\), \(0<a\in\mathbb{R}\), \(0<b\in\mathbb{R}\), \( t\in\mathbb{R}\). We prove that \( S_p(a,b;t)\) is completely monotonic on \((0,\infty)\) as a function of \(a\) (if \(t>0\)) and as a function of \(b\) (if \(t\geq -1)\). This extends a result of Sofo, who proved that \(a\mapsto S_p(a,b;t)\) is strictly decreasing, convex, and log-convex on \([1,\infty)\).

Cantor sets in \(\mathbb{R}\) are common examples of sets for which Hausdorff measures can be positive and finite. However, there exist Cantor sets for which no Hausdorff measure is supported and finite. The purpose of this paper is to try to resolve this problem by studying an extension of the Hausdorff measures \( \mu_h\) on \(\mathbb{R}\), allowing gauge functions to depend on the midpoint of the covering intervals instead of only on the diameter. As a main result, a theorem about the Hausdorff measure of any regular enough Cantor set, with respect to a chosen gauge function, is obtained.

The purpose of this paper is to find necessary and sufficient conditions on the weight \(w\) for the weak type \(\left( p,p\right) \) with \(1\leq p<+\infty \) and for the strong type \(\left( p,p\right)\) with \(1<p<+\infty \) respect to a measure \(w\,dx\) of the vector-valued one-sided maximal operator \(\left(M_{g}^{+}\right) _{r}\).

Riemann, Riemann-Dunford, Riemann-Pettis and Darboux integrable functions with values in a Banach space and Riemann-Gelfand integrable functions with values in the dual of a Banach space are studied in the light of the work of Graves, Alexiewicz and Orlicz, and Gordon. Various properties of these types of integrals and the interrelation between them are established. The Fundamental Theorem of Integral Calculus for these types of integrals is also studied.

We offer a new and simpler proof of a recent \(\epsilon\)-\(\delta\) characterization of Baire class one functions using a theorem by Henri Lebesgue. The proof is more elementary in the sense that it does not use the Baire Category Theorem. Furthermore, the proof requires only that the domain and range be separable metric spaces instead of Polish spaces.

In this paper, we extend Hake’s theorem over metric measure spaces. We provide its measure theoretic versions in terms of the Henstock variational measure \(V_F\).

Three classical constructions of Lebesgue nonmeasurable sets on the real line \(\mathbb{R}\) are envisaged from the point of view of the thickness of those sets. It is also shown, within \({\bf ZF}~\&~{\bf DC}\) theory, that the existence of a Lebesgue nonmeasurable subset of \(\mathbb{R}\) implies the existence of a partition of \(\mathbb{R}\) into continuum many thick sets.

The strong derivative is not, without some caution, a useful tool in the study of McShaneʼs (i.e., Lebesgueʼs) integral. Even so, the underlying structure of that process of derivation is closely connected to the formulation of the Riemann sums definition that McShane gave for his integral. This article discusses some of the features and traps for the study of those connections.