Convergence Theorems in the Sense of Vitali and Lusin for the Itô–Henstock Integral

In this paper, we formulate a version of convergence theorem using double Lusin condition and a version of Vitali convergence theorem for the Itô–Henstock integral of an operator-valued stochastic process with respect to a [Formula: see text]-Wiener process.

The Schrödinger Equation, Path Integration and Applications

The Schrödinger equation is fundamental in quantum mechanics as it makes it possible to determine the wave function from energies and to use this function in the mean calculation of variables, for example, as the most likely position of a group of one or more massive particles. In this paper, we present a survey on some theories involving the Schrödinger equation and the Feynman path integral. We also consider a Feynman–Kac-type formula, as introduced by Patrick Muldowney, with the Henstock integral in the description of the expectation of random walks of a particle. It is well known that the non-absolute integral defined by R. Henstock “fixes” the defects of the Feynman integral. Possible applications where the potential in the Schrödinger equation can be highly oscillating, discontinuous or delayed are mentioned in the end of the paper.

On Measurability of C[0, 1]: Countable Dense Set Approach

In V. Boonpogkrong, On measurability of [Formula: see text], Thai J. Math. (accepted) we have proved that [Formula: see text] is measurable in a setting of a dyadic Henstock integral, where dimension sets are subsets of the dyadic rational. In this paper, we replace “dyadic rational” by “countable dense set” and use a setting of Henstock integral, where dimension sets are subsets of this fixed countable dense set. The collection of measurable sets in this new setting is smaller than that in the old setting. We still can prove that [Formula: see text] is measurable.

On the Differentiation of Henstock and McShane Integrals

It is well known that the derivative of the primitive of 1-dimensional Henstock integral exists almost everywhere. Point-interval pairs used in the derivative are Henstock point-interval pairs, which are consistent with point-interval pairs used in the Henstock integral. Note that “almost everywhere” is a set of points, more precisely, the derivative does not exist on a set of points with measure zero. We can transform a set of Henstock point-interval pairs to a set of points with measure zero because of Vitali’s covering theorem. For 1-dimensional McShane integrals, [Formula: see text]-dimensional McShane and Henstock integrals, covering theorems of Vitali’s type cannot be applied. In this paper, we shall discuss differentiation of [Formula: see text]-dimensional McShane and Henstock integrals.

A version of Hake’s theorem for Kurzweil–Henstock integral in terms of variational measure

We introduce the notion of variational measure with respect to a derivation basis in a topological measure space and consider a Kurzweil–Henstock-type integral related to this basis. We prove a version of Hake’s theorem in terms of a variational measure.

Backwards Ito-Henstock Integral for the Hilbert-Schmidt-Valued Stochastic Process

In this paper, a definition of backwards Ito-Henstock integral for the Hilbert-Schmidt-valued stochastic process is introduced. We formulate the Ito isometry for this integral. Moreover, an equivalent definition for this integral is given using the concept of AC^2 [0,T]-property, a version of absolute continuity.