A Cauchy type condition and integrability
A converse to some inequalities and approximations in the theory of Stieltjes and stochastic integrals, and for nth derivatives
A general form of the product integral and linear ordinary differential equations
A Korovkin-type theorem in the space of Riemann integrable funciotns.
A linear functional differential equation with distributions in the input.
A non absolutely convergent integral which admits transformation and can be used for integration on manifolds
A non-absolutely convergent integral in and the theorem of Gauss
A Precise Vitali Theorem for Lebesgue Measure.
A Radon-Nikodym derivative for positive linear functionals
An exact Radon-Nikodym derivative is obtained for a pair (I,J) of positive linear functionals, with J absolutely continuous with respect to I, using a notion of exhaustion of I on elements of a function algebra lattice.
A representation theorem for operators on a space of interval functions.
A sharp bound of the Čebyšev functional for the Riemann-Stieltjes integral and applications.
Abstract Riemann integrability and measurability
We prove that the spectral sets of any positive abstract Riemann integrable function are measurable but (at most) a countable amount of them. In addition, the integral of such a function can be computed as an improper classical Riemann integral of the measures of its spectral sets under some weak continuity conditions which in fact characterize the integral representation.
Algunas consecuencias elementales del teorema de Weierstrass.
Almost Everywhere First-Return Recovery
We present a new characterization of Lebesgue measurable functions; namely, a function f:[0,1]→ ℝ is measurable if and only if it is first-return recoverable almost everywhere. This result is established by demonstrating a connection between almost everywhere first-return recovery and a first-return process for yielding the integral of a measurable function.
An elementary proof of the theorem that absolute gauge integrability implies Lebesgue integrability
It is commonly known that absolute gauge integrability, or Henstock-Kurzweil (H-K) integrability implies Lebesgue integrability. In this article, we are going to present another proof of that fact which utilizes the basic definitions and properties of the Lebesgue and H-K integrals.
An Elementary Treatise on the Integral Calculus [Book]
An extension of the Riemann-Lebesgue lemma and some applications.
An Integral in Topological Spaces II.
Approximation of Lebesgue Integrals by Riemann Sums and Lattice Points in Domains with Fractal Boundary.
Bochner product integration
A new definition of the product integral is given. The definition is based on a procedure which is analogous to the sum definition of the Bochner integral given by J. Kurzweil and E.J. McShane. The new definition is shown to be equivalent to the seemingly verey different one given by J.D. Dollard and C.N. Friedman in [1] and [2].