Measurable functions similar to the itô integral and the Paley-Wiener-Zygmund integral over continuous paths

Abstract

Let C[0,T] denote an analogue of generalizedWiener space, the space of continuous real-valued functions on the interval [0,T]. On the space C[0,T], we introduce a finite measure w?,?,? and investigate its properties, where ? is an arbitrary finite measure on the Borel class of R. Using the measure w?,?,?, we also introduce two measurable functions on C[0,T], one of them is similar to the It? integral and the other is similar to the Paley-Wiener-Zygmund integral. We will prove that if ?(R) = 1, then w?,?,? is a probability measure with the mean function ? and the variance function ?, and the two measurable functions are reduced to the Paley-Wiener-Zygmund integral on the analogue ofWiener space C[0,T]. As an application of the integrals, we derive a generalized Paley-Wiener-Zygmund theorem which is useful to calculate generalized Wiener integrals on C[0,T]. Throughout this paper, we will recognize that the generalized It? integral is more general than the generalized Paley-Wiener-Zygmund integral.