Measurable functions similar to the It\^o integral and the Paley-Wiener-Zygmund integral over continuous paths

Abstract

Let $C[0,T]$ denote an analogue of generalized Wiener space, the space of continuous real-valued functions on the interval $[0,T]$. On the space $C[0,T]$, we introduce a finite measure $w_{\alpha,\beta;\varphi}$ and investigate its properties, where $\varphi$ is an arbitrary finite measure on the Borel class of $\mathbb R$. Using the measure $w_{\alpha,\beta;\varphi}$, we also introduce two measurable functions on $C[0,T]$; one of them is similar to the It\^o integral and the other is similar to the Paley-Wiener-Zygmund integral. We will prove that if $\varphi(\mathbb R)=1$, then $w_{\alpha,\beta;\varphi}$ is a probability measure with the mean function $\alpha$ and the variance function $\beta$, and the two measurable functions are reduced to the Paley-Wiener-Zygmund integral on the analogue of Wiener 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\^o integral is more general than the generalized Paley-Wiener-Zygmund integral.