Integration

The concept of an integral on a general measure space is developed from first principles. Riemann–Stieltjes and Lebesgue–Stieltjes integrals are defined. The monotone convergence theorem, fundamental properties of integrals, and related inequalities are covered. Other topics include product measure and multiple integrals, Fubini’s theorem, signed measures, and the Radon–Nikodym theorem.

The Family of Central Cantor Sets with Packing Dimension Zero

As in the recent article of M. Balcerzak, T. Filipczak and P. Nowakowski, we identify the family CS of central Cantor subsets of [0, 1] with the Polish space X : = (0, 1)ℕ equipped with the probability product measure µ. We investigate the size of the family P 0 of sets in CS with packing dimension zero. We show that P 0 is meager and of µ measure zero while it is treated as the corresponding subset of X. We also check possible inclusions between P 0 and other subfamilies CS consisting of small sets.

Construction of Fuzzy Measures over Product Spaces

In this paper, we study the problem of constructing a fuzzy measure over a product space when fuzzy measures over the marginal spaces are available. We propose a definition of independence of fuzzy measures and introduce different ways of constructing product measures, analyzing their properties. We derive bounds for the measure on the product space and show that it is possible to construct a single product measure when the marginal measures are capacities of order 2. We also study the combination of real functions over the marginal spaces in order to produce a joint function over the product space, compatible with the concept of marginalization, paving the way for the definition of statistical indices based on fuzzy measures.

Families of symmetric Cantor sets from the category and measure viewpoints

We consider the family {\mathcal{CS}} of symmetric Cantor subsets of {[0,1]} . Each set in {\mathcal{CS}} is uniquely determined by a sequence {a=(a_{n})} belonging to the Polish space {X\mathrel{\mathop{:}}=(0,1)^{\mathbb{N}}} equipped with probability product measure μ. This yields a one-to-one correspondence between sets in {\mathcal{CS}} and sequences in X. If {\mathcal{A}\subset\mathcal{CS}} , the corresponding subset of X is denoted by {\mathcal{A}^{\ast}} . We study the subfamilies {\mathcal{H}_{0}} , {\mathcal{SP}} and {\mathcal{M}} of {\mathcal{CS}} , consisting (respectively) of sets with Haudsdorff dimension 0, and of strongly porous and microscopic sets. We have {\mathcal{M}\subset\mathcal{H}_{0}\subset\mathcal{SP}} , and these inclusions are proper. We prove that the sets {\mathcal{M}^{\ast}} , {\mathcal{H}_{0}^{\ast}} , {\mathcal{SP}^{\ast}} are residual in X, and {\mu(\mathcal{H}_{0}^{\ast})=0} , {\mu(\mathcal{SP}^{\ast})=1} .