Sur une intégrale définie.
Sur une intégrale double
Sur une méthode de variation des paramètres dans les intégrales indéfinies
Sur une question de minimum
The AP-Denjoy and AP-Henstock integrals revisited
The note is related to a recently published paper J. M. Park, J. J. Oh, C.-G. Park, D. H. Lee: The AP-Denjoy and AP-Henstock integrals. Czech. Math. J. 57 (2007), 689–696, which concerns a descriptive characterization of the approximate Kurzweil-Henstock integral. We bring to attention known results which are stronger than those contained in the aforementioned work. We show that some of them can be formulated in terms of a derivation basis defined by a local system of which the approximate basis...
The Basic Existence Theorem of Riemann-Stieltjes Integral
In this article, the basic existence theorem of Riemann-Stieltjes integral is formalized. This theorem states that if f is a continuous function and ρ is a function of bounded variation in a closed interval of real line, f is Riemann-Stieltjes integrable with respect to ρ. In the first section, basic properties of real finite sequences are formalized as preliminaries. In the second section, we formalized the existence theorem of the Riemann-Stieltjes integral. These formalizations are based on [15],...
The Denjoy extension of the Riemann and McShane integrals
In this paper we study the Denjoy-Riemann and Denjoy-McShane integrals of functions mapping an interval into a Banach space It is shown that a Denjoy-Bochner integrable function on is Denjoy-Riemann integrable on , that a Denjoy-Riemann integrable function on is Denjoy-McShane integrable on and that a Denjoy-McShane integrable function on is Denjoy-Pettis integrable on In addition, it is shown that for spaces that do not contain a copy of , a measurable Denjoy-McShane integrable...
The dual of the space of functions of bounded variation
In the paper, we show that the space of functions of bounded variation and the space of regulated functions are, in some sense, the dual space of each other, involving the Henstock-Kurzweil-Stieltjes integral.
The fundamental theorem of calculus for Lebesgue integral.
The hardy-littlewood maximal function and derivation of integrals.
The Henstock-Kurzweil integral
We present a method of integration along the lines of the Henstock-Kurzweil integral. All -derivatives are integrable in this method.
The Lebesgue integral a little over 100 years later.
The Linearity of Riemann Integral on Functions from ℝ into Real Banach Space
In this article, we described basic properties of Riemann integral on functions from R into Real Banach Space. We proved mainly the linearity of integral operator about the integral of continuous functions on closed interval of the set of real numbers. These theorems were based on the article [10] and we referred to the former articles about Riemann integral. We applied definitions and theorems introduced in the article [9] and the article [11] to the proof. Using the definition of the article [10],...
The Measurability of Complex-Valued Functional Sequences
In this article, we formalized the measurability of complex-valued functional sequences. First, we proved the measurability of the limits of real-valued functional sequences. Next, we defined complex-valued functional sequences dividing real part into imaginary part. Then using the former theorems, we proved the measurability of each part. Lastly, we proved the measurability of the limits of complex-valued functional sequences. We also showed several properties of complex-valued measurable functions....
The monotone convergence theorem for multidimensional abstract Kurzweil vector integrals
We prove two versions of the Monotone Convergence Theorem for the vector integral of Kurzweil, , where is a compact interval of , and are functions with values on and respectively, and and are monotone ordered normed spaces. Analogous results can be obtained for the Kurzweil vector integral, , as well as to unbounded intervals .
The space of Henstock integrable functions of two variables.
The surface integral
Transformation of -dimensional Lebesgue integrals
Über eine Formel für numerische Berechnung der bestimmten Integrale
Ueber den Werth einiger Integrale.