product measure
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2021 ◽  
pp. 80-102
Author(s):  
James Davidson

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.



2021 ◽  
Vol 78 (1) ◽  
pp. 1-8
Author(s):  
Piotr Nowakowski

Abstract 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.



2021 ◽  
Vol 31 (1) ◽  
pp. 425-447
Author(s):  
Jérôme Stenger ◽  
Fabrice Gamboa ◽  
Merlin Keller


2020 ◽  
Vol 491 (1) ◽  
pp. 124272
Author(s):  
Jian-Yong Wang ◽  
Han Yu


Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1605 ◽  
Author(s):  
Fernando Reche ◽  
María Morales ◽  
Antonio Salmerón

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.



2020 ◽  
Author(s):  
Hamide Ranjbar Shamsi ◽  
Maryam Omidi Najafabadi ◽  
Seyed Jamal Farajollah Hosseini


2019 ◽  
Vol 26 (4) ◽  
pp. 545-553
Author(s):  
Marek Balcerzak ◽  
Tomasz Filipczak ◽  
Piotr Nowakowski

Abstract 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} .



Author(s):  
Satish Shirali ◽  
Harkrishan Lal Vasudeva
Keyword(s):  


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