This obituary in memory of Washek F. Pfeffer describes his life and character. It also touches on his work in topological measure theory, but other aspects of his research are treated in the adjoining articles by De Pauw, Gruenhage, and Moonens.

Hereunder, I describe some contributions of Washek F. Pfeffer and his collaborators, to integration theory, starting from his remarkable paper “The Gauss-Green Theorem,” published in 1991.

We here briefly describe the importance of Washek F. Pfeffer’s three books on Riemann-type integration and its applications.

Washek Pfeffer's contributions in general topology are discussed, especially his paper with van Douwen on $D$-spaces and his work withPrikry on small spaces.

It is humbling to write an obituary for Togo Nishiura, knowing that I can but touch on so few of the aspects of such a fully lived and loved life. He was a complex man who lived a simple life devoted to his mathematics, beliefs, family and friendships. I first met him as an undergraduate at the University of Wisconsin-Milwaukee and he has encouraged and supported me since I was a graduate student and the activities of the Real Analysis Exchange since its inception.

In spite of the Lebesgue density theorem, there is a positive $\delta$ such that, for every measurable set $A\subset \mathrm{ℝ}$ with and , there is a point at which both the lower densities of $A$ and of the complement of A are at least $\delta$. The problem of determining the supremum ${\delta }_{\mathcal{H}}$ of possible values of this $\delta$ was studied by V. I. Kolyada, A. Szenes and others, and it was solved by O. Kurka. Lower density of $A$ at $x$ is defined as a lower limit of $\lambda (A\cap [x-h,x+h\left]\right)/2h$. Replacing $\lambda (A\cap [x-h,x+h\left]\right)/2h$ by $\lambda (A\cap [x-t_n,x+t_n\left]\right)/2t_n$ for a fixed decreasing sequence $⟨t⟩$ tending to zero, we obtain a definition of the constant ${\delta }_{⟨t⟩}$. In our paper we look for an upper bound of all such constants.

There are three main contributions in this work. First, the proof that every stabilized asymptotic-${\ell }_1$ Banach space has the Property of Lebesgue is generalized to the coordinate-free case. Second, the proof that every Banach space with the Property of Lebesgue has a unique ${\ell }_1$ spreading model is generalized to cover a particular class of asymptotic models. Third, a characterization of the Property of Lebesgue is derived that applies to those Banach spaces with bases that admit in a strong sense favorable block bases. These results are significant because they demonstrate not only the efficacy of characterizing the Property of Lebesgue in terms of a connection between the local and the global asymptotic structures of certain Banach spaces, but also the possibility of finding a more general characterization in similar terms.

Let $X$ be a complete positive $\sigma$–finite measure space and $\{A_t{\}}_{t\ge 0}$ be a symmetric diffusion semigroup of contraction operators on $L_p\left(X\right)$. We prove that for , the domain of the infinitesimal generator of the semigroup is precisely the space $\left\{{\int }_0^1{A}_{s}h ds:h\in L_p\left(X\right)\right\}$. We also establish that for , the function spaces

$$\left\{2^{n-1}{\int }_0^{2-(n-1)}{A}_{s}h ds|h\in L_p(X)\right\}$$

are equal for every $n\in \mathrm{\mathbb{Z}}$.

Let

$${\mu }_n(\alpha ,\beta ;p,x)=\frac{\left(A_n\right(p,x){)}^{\alpha }}{\left(G_n\right(p,x){)}^{\beta }}(n\ge 2;\alpha ,\beta \in \mathrm{ℝ}),$$

where $A_n(p,x)$ and $G_n(p,x)$ denote the arithmetic and geometric means of $x=(x_1,\dots ,x_n)\in {\mathrm{ℝ}}_{+}^n$ with weights $p=(p_1,\dots ,p_n)\in {\mathrm{ℝ}}_{+}^n$, $p_1+\cdots +p_n=1$. We prove:

(i) The inequality

$${\mu }_n(\alpha ,\beta ;p,x+y)\le {\mu }_n(\alpha ,\beta ;p,x)+{\mu }_n(\alpha ,\beta ;p,y)(*)$$

is valid for all $x,y\in {\mathrm{ℝ}}_{+}^n$ and $p\in {\mathrm{ℝ}}_{+}^n$ with $p_1+\cdots +p_n=1$ if and only if $\beta \ge \max (\alpha -1,0)$.

(ii) Inequality $(*)$ with “$\ge$” instead of “$\le$” holds for all $x,y\in {\mathrm{ℝ}}_{+}^n$ and $p\in {\mathrm{ℝ}}_{+}^n$ with $p_1+\cdots +p_n=1$ if and only if $\alpha \ge 0$ and $\beta \le \max (\alpha -1,0)$.

This extends a result of Dragomir, Comănescu and Pearce, who proved $(*)$ for the special case $\alpha =2$, $\beta =1$.

We introduce an alternative definition of the concept of an ideal weak QN-space and compare it with the definition introduced by Bukovský, Das, and Šupina. We classify the properties of spaces expressing some kinds of indistinguishability for various pairs of ideal convergences and semi-convergences. We give combinatorial characterizations of the least cardinalities of spaces not having a particular property and show that they are invariant for classes of spaces that contain metric spaces and are closed under homeomorphisms. The counterexamples proving this are subsets of the Baire space ${}^{\omega }\omega$.

In [13] we gave combinatorial characterizations of $\mathrm{non}\left(P\right)$ of spaces expressing non-distinguishability of some ideal convergences and semi-convergences of sequences of continuous functions. In the present paper we study three of these invariants: $\mathrm{non}\left(\right(I,JQN)-\mathrm{space})$, $\mathrm{none}\left(\right(I,{\le }_{\mathrm{K}}JQN)-\mathrm{space})$, and $\mathrm{none}\left(\mathrm{w}\right(I,JQN)-\mathrm{space})$. We study them in connection with partial orderings of ${}^{\omega }\omega$ restricted to relations between $I$-to-one functions and $J$-to-one functions. In particular we prove that $\mathrm{none}\left(\mathrm{w}\right(I,JQN)-\mathrm{space})\le \mathfrak{b}$ for every capacitous ideal $J$ on $\omega$. This generalizes the same result of Kwela for ideals $J$ contained in an $F_{\sigma }$-ideal. If $J$ is a capacitous $P$-ideal, then $\mathrm{non}\left(\right(I,JQN)-\mathrm{space})=\mathrm{none}\left(\right(I,{\le }_{\mathrm{K}}JQN)-\mathrm{space})=\mathfrak{b}$ for every ideal $I\subseteq J$ and $\mathrm{none}\left(\mathrm{w}\right(I,JQN)-\mathrm{space})=\mathfrak{b}$ for every ideal $I$ below $J$ in the Katĕtov partial quasi-ordering of ideals.

We show that a few basic classes of lower semicontinuous functions on ${\mathrm{ℝ}}^n$ are densely recoverable. Specifically, we show that the sum of a convex and a continuous function, the difference of two convex and lower semicontinuous functions, a K-increasing function (where K is a cone of nonempty interior), and differences of K-increasing functions are all functions uniquely determined by their values on a dense set in ${\mathrm{ℝ}}^n$. Thus, sets of such functions of each type are densely recoverable sets. In general, the sum and difference of two densely recoverable sets of functions is shown to not be densely recoverable.

We prove a stronger version of a conjecture stated in a paper from 2017 by J. M. Ash and S. Catoiu concerning relations between various notions of the Lipschitz property and differentiability in the Euclidean plane. We also provide an improved version of the main result of that paper.

The basic goal of quantization for probability distribution is to reduce the number of values, which is typically uncountable, describing a probability distribution to some finite set and thus approximation of a continuous probability distribution by a discrete distribution. Mixed distributions are an exciting new area for optimal quantization. In this paper, we have determined the optimal sets of $n$-means, the $n$th quantization errors, and the quantization dimensions of different mixed distributions. Besides, we have discussed whether the quantization coefficients for the mixed distributions exist. The results in this paper will give a motivation and insight into more general problems in quantization for mixed distributions.

In this paper we present a proof of the following statement: there is a ${\mathcal{C}}^{\infty }$ function $f$ on ${\mathrm{ℝ}}^n$ such that any other ${\mathcal{C}}^{\infty }$ function on ${\mathrm{ℝ}}^n$ is the uniform limit, on the compact subsets of ${\mathrm{ℝ}}^n$, of translations of $f$ by natural numbers. This is a real version of the well-known Birkhoff’s result on the existence of a function with a similar property in the space of entire functions. Afterwards, we show that the technique used in our proof allows us to create $2^{{\aleph }_0}$ linearly independent real ${\mathcal{C}}^{\infty }$ universal functions. We also demonstrate that we may even obtain real analytic universal functions (in the sense of translations) by using Whitney’s Approximation Theorem.

The paper presents a generalization of the density point’s notion to the ideal-convergence framework. For an ideal $\mathcal{I}\subseteq \mathcal{P}\left(\mathrm{\mathbb{N}}\right)$ (with $\mathrm{Fin}\subseteq \mathcal{I}$), Lebesgue measurable set $A\subseteq \mathrm{ℝ}$ we introduce a definition of a density point of A with respect to $\mathcal{I}$; we prove that the classical approach fits into this generalization (Theorem 4); we construct a family of Cantorlike sets showing that Lebesgue Density Theorem cannot be maximally improved in this direction (Theorem 8).

A function $F:[a,b]\to X$ is said to be an $\mathrm{SL}$ function if it satisfies the Strong Lusin $\left(SL\right)$ condition given as follows: for every $\theta$-nbd $U$ and a set $E\subset [a,b]$ of measure zero, there exists a gauge $\delta$ such that for every $\delta$-fine partial partition $D=\left\{\right([x_{i-1},x_i],t_i):1\le i\le n\}$ of $[a,b]$ with $t_i\in E$, there exist $\theta$-nbds $U_1,U_2,\dots ,U_n$ such that $\sum _{i=1}^n{U}_i\subseteq V$ and $F\left(x_i\right)-F\left(x_{i-1}\right)\in U_i$ for each $i=1,2,\dots ,n$. In this paper, we introduce the SL integral of a function taking values on a locally convex topological vector space (LCTVS). Further, we show that this integral is equivalent to a stronger version of the Henstock integral.

Let $p_n$ denote the $n$th prime, and consider the function $1/n\mapsto 1/p_n$ which maps the reciprocals of the positive integers bijectively to the reciprocals of the primes. We show that Hölder continuity of this function is equivalent to a parametrised family of Cramér type estimates on the gaps between successive primes. Here the parametrisation comes from the Hölder exponent. In particular, we show that Cramér’s conjecture is equivalent to the map $1/n\mapsto 1/p_n$ being Lipschitz. On the other hand, we show that the inverse map $1/p_n\mapsto 1/n$ is Hölder of all orders but not Lipschitz and this is independent of Cramér’s conjecture.