We investigate Baire classes of strongly affine mappings with values in Fréchet spaces. We show, in particular, that the validity of the vector-valued Mokobodzki result on affine functions of the first Baire class is related to the approximation property of the range space. We further extend several results known for scalar functions on Choquet simplices or on dual balls of L₁-preduals to the vector-valued case. This concerns, in particular, affine classes of strongly affine Baire mappings, the...

Let $X$ be a complex $L_1$-predual, non-separable in general. We investigate extendability of complex-valued bounded homogeneous Baire-$\alpha$ functions on the set $\mathrm{ext}{B}_{X^{*}}$ of the extreme points of the dual unit ball $B_{X^{*}}$ to the whole unit ball $B_{X^{*}}$. As a corollary we show that, given $\alpha \in [1,{\omega }_1)$, the intrinsic $\alpha$-th Baire class of $X$ can be identified with the space of bounded homogeneous Baire-$\alpha$ functions on the set $\mathrm{ext}{B}_{X^{*}}$ when $\mathrm{ext}{B}_{X^{*}}$ satisfies certain topological assumptions. The paper is intended to be a complex counterpart to the same authors’...

A characterization of functions in the first Baire class in terms of their sets of discontinuity is given. More precisely, a function $f:ℝ\to ℝ$ is of the first Baire class if and only if for each $ϵ>0$ there is a sequence of closed sets ${\left\{C_n\right\}}_{n=1}^{\infty }$ such that $D_f={\bigcup }_{n=1}^{\infty }{C}_n$ and ${\omega }_f\left(C_n\right)<ϵ$ for each $n$ where $${\omega }_f\left(C_n\right)=sup\left\{\right|f\left(x\right)-f\left(y\right)|:x,y\in C_n\}$$ and $D_f$ denotes the set of points of discontinuity of $f$. The proof of the main theorem is based on a recent $ϵ$-$\delta$ characterization of Baire class one functions as well as on a well-known theorem due to Lebesgue. Some direct applications of...

We investigate Baire-one functions whose graph is contained in the graph of a usco mapping. We prove in particular that such a function defined on a metric space with values in ${ℝ}^d$ is the pointwise limit of a sequence of continuous functions with graphs contained in the graph of a common usco map.

We prove that any Baire-one usco-bounded function from a metric space to a closed convex subset of a Banach space is the pointwise limit of a usco-bounded sequence of continuous functions.

It is shown that a Banach-valued Henstock-Kurzweil integrable function on an $m$-dimensional compact interval is McShane integrable on a portion of the interval. As a consequence, there exist a non-Perron integrable function $f{[0,1]}^2⟶ℝ$ and a continuous function $F{[0,1]}^2⟶ℝ$ such that $$\left(\P \right){\int }_0^x\left\{\left(\P \right){\int }_0^{y}f(u,v)\mathrm{d}v\right\}\mathrm{d}u=\left(\P \right){\int }_0^y\left\{\left(\P \right){\int }_0^{x}f(u,v)\mathrm{d}u\right\}\mathrm{d}v=F(x,y)$$ for all $(x,y)\in {[0,1]}^2$.

A topological space $X$ is called base-base paracompact (John E. Porter) if it has an open base $ℬ$ such that every base ${ℬ}^{\text{'}}\subseteq ℬ$ has a locally finite subcover $𝒞\subseteq {ℬ}^{\text{'}}$. It is not known if every paracompact space is base-base paracompact. We study subspaces of the Sorgenfrey line (e.g. the irrationals, a Bernstein set) as a possible counterexample.

We consider real valued functions $f$ defined on a subinterval $I$ of the positive real axis and prove that if all of $f$’s quantum differences are nonnegative then $f$ has a power series representation on $I$. Further, if the quantum differences have fixed sign on $I$ then $f$ is analytic on $I$.

Let $E$ be a self-similar set with similarities ratio $r_j(0\le j\le m-1)$ and Hausdorff dimension $s$, let $\overrightarrow{p}(p_0,p_1)...p_{m-1}$ be a probability vector. The Besicovitch-type subset of $E$ is defined as$$E\left(\overrightarrow{p}\right)=\left\{x\in E:\underset{n\to \infty }{lim}\frac{1}{n}\sum _{k=1}^n{\chi }_j\left(x_k\right)=p_j,0\le j\le m-1\right\},$$where ${\chi }_j$ is the indicator function of the set $\left\{j\right\}$. Let $\alpha ={dim}_H\left(E\left(\overrightarrow{p}\right)\right)={dim}_P\left(E\left(\overrightarrow{p}\right)\right)=\frac{{\sum }_{j=0}^{m-1}{p}_{j}log{p}_j}{{\sum }_{j=0}^{m-1}{p}_{i}log{r}_j}$ and $g$ be a gauge function, then we prove in this paper:(i) If $\overrightarrow{p}=(r_0^s,r_1^s,\cdots ,r_{m-1}^s)$, then$${\mathscr{H}}^s\left(E\left(\overrightarrow{p}\right)\right)={\mathscr{H}}^s\left(E\right),{𝒫}^s\left(E\left(\overrightarrow{p}\right)\right)={𝒫}^s\left(E\right),$$moreover both of ${\mathscr{H}}^s\left(E\right)$ and ${𝒫}^s\left(E\right)$ are finite positive;(ii) If $\overrightarrow{p}$ is a positive probability vector other than $(r_0^s,r_1^s,\cdots ,r_{m-1}^s)$, then the gauge functions can be partitioned as follows$${\mathscr{H}}^g\left(E\left(\overrightarrow{p}\right)\right)=+\infty \iff \underset{t\to 0}{\overline{\lim }}\frac{logg\left(t\right)}{logt}\le \alpha ;{\mathscr{H}}^g\left(E\left(\overrightarrow{p}\right)\right)=0⟺\underset{t\to 0}{\overline{\lim }}\frac{logg\left(t\right)}{logt}\&gt;\alpha ,$$$$...$$