Let $X$ be a metric space and let $\mu$ be a Borel measure on $X$. We say that $\mu$ satisfies the strong Vitali property if for any Borel subset $E$ of $X$ with $\mu (E) <\infty$ and for any fine cover $\mathcal{V}$ of $E$, we may extract a countable disjoint subcollection $\pi = \{B_{i}\}$ from $\mathcal{V}$ such that $\displaystyle \mu(E \setminus \cup B_{i}) = 0$. If we require $\pi$ to satisfy the condition that if $B(x_{i},r_{i}), ~B(x_{j},r_{j}) \in \pi$ with $i \neq j$, then $x_{i} \notin B(x_{j},r_{j})$, then $\mu$ is said to satisfy the weak Vitali property. Besicovitch showed that every finite Borel measure in $\mathbb{R}^{n}$ must satisfy the strong Vitali property. It is also true for certain other metric spaces. In a general metric space it is no longer pertinent to ask if all Borel measures possess a certain property. To this end we construct a metric space $\Omega$ and identify two subsets $A, B \subseteq \Omega$ such that for any Borel probability measure $\mu$ on $\Omega$,

1.$\mu (A) = 1$, then $\mu$ must satisfy the strong Vitali property;

2. $\mu(B) = 1$, then $\mu$ must satisfy the weak Vitali property but not necessarily the strong Vitali property;

We introduce the notion of a centralized Vitali property and give an example of a measure for which this property fails.

A real function is called strongly $\mathbb{Q}$-dif\-fer\-en\-ti\-able if, for every real number $ h \,$, the limit of the ratio $ \left( f(x+rh) - f(x) \right) / r $ exists whenever $x$ tends to any fixed real number and $r$ tends to zero through the positive rationals. After examining the dependence of strong $\mathbb{Q}$-derivatives on their parameters, we prove that every strongly $\mathbb{Q}$-dif\-fer\-en\-ti\-able function can be represented as the sum of an additive mapping and a continuously dif\-fer\-en\-ti\-able function.

We show which subspaces of Darboux-like real function spaces are porous or boundary sets with the metric of uniform convergence.

Given a power series $f(x)=\sum_{n=1}^\infty\,a_n\,x^n$ with nonnegative coefficients satisfying $\sum_{n=1}^\infty\,a_n=1$ we give sufficient conditions on the sequence $(a_n)$ to guarantee that the coefficients of the Taylor series of $h(x)=1/(1-f(x))$ form a nonincreasing sequence. This type of result is useful when one wishes to apply Tauberian theorems.

We consider the set of Baire 1 functions endowed with the pointwise partial ordering and investigate the structure of the linearly ordered subsets.

We construct a doubling gauge function $g$ and a compact set $L\subset \mathbb{R}$ for which $\mathcal{P}^g(L)<\mathcal{P}_0^g(L)<\infty$

For a given commutative group $G$ equipped with a nonzero $\sigma$-finite diffused measure $\mu$, the question of the existence of a subgroup of $G$, nonmeasurable with respect to $\mu$, is discussed.

We define a complete metric on $C^\infty$, and find that most functions in $C^\infty$ are nowhere Gevrey differentiable of any order. For any $s > 1$ we prove there exists an everywhere Gevrey differentiable function of order $s$ that is nowhere Gevrey differentiable of any order less than $s$.

We present here a unified theory of bilateral derivates, which we call here briefly "biderivates". This unified theory is achieved with the help of two new fundamental theorems on biderivates, called the First and Second Biderivate Theorems. These two biderivate theorems are obtained in terms of bimonotonicity and bi-Lipschitz properties of a function on a set which also depend on the values of the function outside the set like the properties $VB_*$ and $AC_*.$ From these two biderivate theorems we further deduce the Third and Fourth Biderivate Theorems, which deal with the properties of a function on a portion of a given set and the Fifth Biderivate Theorem on the Baire class of biderivates. Next, given $f:X\to\mathbb{R},$ where $X\subset \mathbb{R},$ let $X^\prime$ denote the set of limit points of $X$ in $X.$ Then we define the ``median'' $Mf$ of $f$ to be the multifunction $Mf(x)= [\underline Df(x),\overline Df(x)],$ $x\in X^\prime.$ We deduce here from the above biderivate theorems five basic median theorems which deal with the properties of the median. The various known results on biderivates are deduced here from these biderivate and median theorems many of which are strengthened in this process. As the known results on biderivates were obtained earlier by ad~hoc methods, the present theory provides a synthesis of these results and brings out their inter-relations leading thereby to a unified theory. In particular, we deduce from one of the median theorems several monotonicity theorems in terms of biderivates including an extended version of the well known Goldowski-Tonelli theorem. Further, we deduce from another median theorem a mean-value theorem in terms of median and derivative, and the Darboux property of median and derivative. Also, the Denjoy property of derivatives is obtained from the Third Biderivate Theorem and results on the Baire class of derivatives and medians are obtained from the Fifth Biderivate Theorem. Also, a biderivate version of the classical Denjoy-Young-Saks theorem is obtained from the First Biderivate Theorem. From the above mentioned biderivate and median theorems we also deduce some other known theorems of Denjoy, Young, Choquet, Zahorski, Kronrod, Fort and Marcus, along with the following two new results: (i)a theorem on biderivates similar to a fundamental theorem of Denjoy on unilateral derivates; and (ii)a theorem on median derivates similar to a theorem of Morse on unilateral derivates.

In this paper we prove that for any modulus of continuity on $[0,\infty)$ there exists a concave majorant that is infinitely differentiable on $(0,\infty)$ and satisfies an additional inequality. This extends the results of Stechkin and Korneychuk obtained previously without the requirement that majorants be differentiable.

We give an affirmative answer to a question of K. Ciesielski by showing that the composition $f\circ g$ of two derivatives $f,g:[0,1]\to[0,1]$ always has a fixed point. Using Maximoff's theorem we obtain that the composition of two $[0,1]\to[0,1]$ Darboux Baire-1 functions must also have a fixed point.

In this paper we investigate problems connected with some generalization of spaces $\cal{A}^{\star}$ and $\cal{Q}_{\cal{D}}$ considered in [3].

In three different problems concerning Dirichlet series, we study in detail the role and optimality of a translation by $1/2$ on the abscissas. In particular, we show the non-existence of Rudin-Shapiro like Dirichlet polynomials.

Baek in [2] and [3] defines the covering measures $h^s$ and $Q^s$ for a perturbed Cantor set $F$. He shows that when $s$ is the dimension of the covering measures $h^{s\text{ }}$and $Q^s$ on $F$, then $s$ is the Hausdorff dimension and the packing measure dimension on $F$. In this paper, it is shown that for the perturbed Cantor set $F$, the Hausdorff measure is equal to the covering measure $h^s$ on $F$. Under more restrictions on the set $F$, the packing measure is equal to $2\cdot Q^s.$ Similar results are shown for the weakly convergent deranged Cantor set.

We consider a one parameter family of self-similar sets of overlapping construction. We study the exceptional set; that is the set of those parameters for which the correlation dimension is smaller than the similarity dimension. We find a connection between the exceptional set and the multifractal analysis of a measure.

It is shown that every cliquish function $f$ mapping a pseudometrizable space $X$ into a separable metric space $Y$ can be expressed as the quasiuniform limit of a sequence of quasicontinuous functions $f_k$.

It is shown that in the space ${\R}^{]0,1]}$ any function which is Lebesgue integrable with respect to Wiener measure is also Henstock integrable.

Let $\mathcal{R}$ be any ring of subsets of a set $X$ which is not an algebra and let $\mathcal{A}$ be the algebra generated by $\mathcal{R}$. Suppose that $\mu$ is a countably additive measure on $\mathcal{R}$ and that $\mu^*$ is the outer measure generated by $(\mu,\mathcal{R})$. If $X$ is a countable union of sets in $\mathcal{R}$, then there is a unique countably additive measure $\nu$ on $\mathcal{A}$ which extends $\mu$, and the outer measure generated by $(\nu,\mathcal{A})$ coincides with $\mu^*$. If $X$ is not a countable union of sets in $\mathcal{R}$, then there exists a family $\{ \mu_p : 0 \leq p \leq \infty \}$ of countably additive measures on $\mathcal{A}$ such that each $\mu_p$ agrees with $\mu$ on $\mathcal{R}$. For $0 \leq p \leq \infty$, let $\mu_p^*$ denote the outer measure generated by $(\mu_p, \mathcal{A})$. Then we have $\mu_0^* \leq \mu_p^* \leq \mu_q^* \leq \mu_\infty^* =\mu^*$ for $0< p < q < \infty$. Moreover, if $\mathcal{M}$ and $\mathcal{M}_p$, respectively, denotes the $\sigma$-algebra of $\mu^*$-measurable and $\mu_p^*$-measurable sets, then $\mathcal{M}_p = \mathcal{M}_1 \subset \mathcal{M}_0 = \mathcal{M}_\infty = \mathcal{M}$ for all positive real numbers $p$. As examples, we give countably additive measures on rings for which $\mathcal{M} = \mathcal{M}_1$ and $\mathcal{M} \neq \mathcal{M}_1$, respectively. By the outer measures generated by $\mu$ we shall mean the outer measures $\mu^*$ and $\mu_p^*$ $(0 \leq p \leq \infty)$.

It is shown that if $f$ is $n$-convex then the four $n$th order Peano derivates of $f$ are respectively equal to the corresponding $n$th order approximate Peano derivates and the porosity Peano derivates of $f$. It is further shown that the same result holds for the de la Vall\'ee Poussin derivates, and the symmetric and unsymmetric Riemann derivates.

Let $ p \ge 1 $ and $ a_1, \dots , a_n $ be positive given numbers. We prove that, the family of all solids of $ {\mathcal R}^n $ of the type $\sum_{i=1}^n \left( \frac{|{x_i}| }{ r^{a_i}} \right)^p < 1 $, $ r > 0 $ satisfies the Besicovitch covering lemma if and only if $ p \ge \frac{\max a_i }{\min a_i } $.

n the paper [2] Josef Tkadlec gave an example of a finite singular Borel measure $\mu$ on the real line such that all $\sigma$-porous sets are of $\mu$-measure zero. We give an alternative proof, i.e. probably a simpler construction, of his theorem [2, theorem] and we also give a similar example in $\mathbb{R}^n.

In this note we deal with some problems related to the $H_1$-integral introduced recently by Garces, Lee, and Zhao. We give a~new definition of that integral and characterize $H_1$ integrable functions almost everywhere equal to zero. We also discuss some results stated in the original paper on the $H_1$-integral.

We examine quasi self-similarity for Cantor measures on Cantor sets. A characterization is obtained in terms of the ratios of dissection of the Cantor set. The multifractal theory of Cantor measures is studied, extending the analysis for quasi self-similar measures.

We give a completely elementary proof of the strongest possible version of the change-of-variables formula for the Riemann integration of an integrable function $f$: \[ \int_{a}^{b}f\circ G\left( x\right) g\left( x\right) \,dx=\int_{G\left( a\right) }^{G\left( b\right) }f\left( u\right) \,du \] whenever $g$ is Riemann integrable over $\left[ a,b\right] $, $G\left( x\right) =G\left( a\right) +\int_{a}^{x}g\left( t\right) \,dt$, and $f$ is Riemann integrable over the image of $\left[ a,b\right] $ under $G$. Our arguments do not require $f$ to be real-valued; it can take its values in an arbitrary Banach space.

Chistyakov has proved ``Helly's selection theorem'' - a uniformly BV sequence has a pointwise convergent subsequence - for Banach-(resp. continuous, group-) valued functions from a real interval into a compact subset. We extend, dispensing with continuity, to arbitrary real subsets and lighten compactness of the range to pointwise precompactness (which answers one of his questions). In addition, we accomplish his selection more generally for complete metric-set-valued BV maps with closed graphs which are pointwise compact on dense subsets of their domains.

We describe a new method of constructing universally null sets and universally meager sets using families of Borel sets well-ordered by inclusion.

We prove that every topologically transitive map $f$ on the real line must satisfy the following properties:

(1) The set $C$ of critical points is unbounded.

(2)The set $f(C)$ of critical values is also unbounded.

(3)Apart from the empty set and the whole set, there can be at most one open invariant set.

(4)With a single possible exception, for every element $x$ the backward orbit $\{y\in {\mathbb R} : f^n(y) = x$ for some $n$ in ${\mathbb N}\}$ is dense in ${\mathbb R}$.

Every analytic real closed proper sub-field of $\mathbb R$ has Hausdorff dimension zero. Equivalently, every analytic set of real numbers having positive Hausdorff dimension contains a transcendence base for $\mathbb R$.

In this note we show that a theorem due to L.I. Holder on the absolute summability of Fourier series by the Bosanquet-Linfoot method $|\alpha,\beta|,$ can be improved upon. Our theorem then provides a better refinement to the classical theorem of Bosanquet on summability $|C,\gamma|$ of Fourier series.

In this paper we compare and contrast two indicators of size in the real line: porosity (porosity index) and thickness. We use a relationship between thickness and the porosities to find a condition on when the sum of two Cantor sets is an interval. Finally, some questions are posed.

Given a continuous function $X(t)$ mapping $[0,1]$ continuously onto $[0,1]$, several properties are given which this function must fulfill if there is to be another continuous function $Y(t)$ so that $(X(t),Y(t))$ takes the unit interval onto the unit square.

The sets of type $H^{(s)}$ form an important subclass of the class of sets of uniqueness ($U$-sets) for trigonometrical series. The main aim of this note is to present a proof of an unpublished result of P. \v Sleich which asserts that each set of type $H^{(s)}$ is $\sigma$-bilaterally porous.

We define operators $\mathcal{E}(\ \cdot \ )$ and $\mathcal{N}(\ \cdot \ )$ for blocking sets for almost continuous functions. Using the language of these operators we give new proofs for some classical theorems and prove some new theorems. Finally, we make some remarks regarding uniform limits of almost continuous functions.

We give an elementary real analysis proof that an analytic function on an open region, not constant on any non-void open subset, is an open mapping.

We give a new elementary proof of the Classical Theorem: Let $f$ be of bounded variation on $[a,b]$ and let $V$ be its total variation function. Then there is a set $N$ such that $m\bigl (V(N)\bigr ) = m\bigl (f(N)\bigr ) = m(N) = 0$, and for each $x$ not in $N$, $f$ and $V$ have derivatives, finite or infinite, and $V^\prime(x) = |f^\prime(x)|$.

The author gives corrected statements of results in [M] and a corrected proof of Theorem 1 of [M].