Displaying similar documents to “The monotone convergence theorem for multidimensional abstract Kurzweil vector integrals”

Generalized M-norms on ordered normed spaces

I. Tzschichholtz, M. R. Weber (2005)

Banach Center Publications

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L-norms and M-norms on Banach lattices, unit-norms and base norms on ordered vector spaces are well known. In this paper m- and m ≤ -norms are introduced on ordered normed spaces. They generalize the notions of the M-norm and the order-unit norm, possess also some interesting properties and can be characterized by means of the constants of reproducibility of cones. In particular, the dual norm of an ordered Banach space with a closed cone turns out to be additive on the dual cone if and...

Cauchy's residue theorem for a class of real valued functions

Branko Sarić (2010)

Czechoslovak Mathematical Journal

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Let [ a , b ] be an interval in ℝ and let F be a real valued function defined at the endpoints of [ a , b ] and with a certain number of discontinuities within [ a , b ] . Assuming F to be differentiable on a set [ a , b ] ∖ E to the derivative f , where E is a subset of [ a , b ] at whose points F can take values ± ∞ or not be defined at all, we adopt the convention that F and f are equal to 0 at all points of E and show that 𝒦ℋ -vt ∫ a b f = F ( b ) - F ( a ) , where 𝒦ℋ -vt denotes the total value of the integral. The paper ends with a few examples that illustrate the...

Role of the Harnack extension principle in the Kurzweil-Stieltjes integral

Umi Mahnuna Hanung (2024)

Mathematica Bohemica

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In the theories of integration and of ordinary differential and integral equations, convergence theorems provide one of the most widely used tools. Since the values of the Kurzweil-Stieltjes integrals over various kinds of bounded intervals having the same infimum and supremum need not coincide, the Harnack extension principle in the Kurzweil-Henstock integral, which is a key step to supply convergence theorems, cannot be easily extended to the Kurzweil-type Stieltjes integrals with...

The L r Henstock-Kurzweil integral

Paul M. Musial, Yoram Sagher (2004)

Studia Mathematica

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We present a method of integration along the lines of the Henstock-Kurzweil integral. All L r -derivatives are integrable in this method.

Continuity in the Alexiewicz norm

Erik Talvila (2006)

Mathematica Bohemica

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If f is a Henstock-Kurzweil integrable function on the real line, the Alexiewicz norm of f is ∥ f ∥ = sup I | ∫ I f | where the supremum is taken over all intervals I ⊂ ℝ . Define the translation τ x by τ x f ( y ) = f ( y - x ) . Then ∥ τ x f - f ∥ tends to 0 as x tends to 0 , i.e., f is continuous in the Alexiewicz norm. For particular functions, ∥ τ x f - f ∥ can tend to 0 arbitrarily slowly. In general, ∥ τ x f - f ∥ ≥ osc f | x | as x → 0 , where osc f is the oscillation of f . It is shown that if F is a primitive of f then ∥ τ x F - F ∥ ≤ ∥ f ∥ | x | . An example shows that the function y ↦ τ x F ( y ) - F ( y ) need not be in L 1 . However, if...

Henstock-Kurzweil integral on BV sets

Jan Malý, Washek Frank Pfeffer (2016)

Mathematica Bohemica

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The generalized Riemann integral of Pfeffer (1991) is defined on all bounded BV subsets of ℝ n , but it is additive only with respect to pairs of disjoint sets whose closures intersect in a set of σ -finite Hausdorff measure of codimension one. Imposing a stronger regularity condition on partitions of BV sets, we define a Riemann-type integral which satisfies the usual additivity condition and extends the integral of Pfeffer. The new integral is lipeomorphism-invariant and closed with respect...