ON THE SL-INTEGRAL OF LCTVS-VALUED FUNCTIONS

Abstract

A function $F:[a,b]\to X$ is said to be an $\mathrm{SL}$ function if it satisfies the Strong Lusin $\left(SL\right)$ condition given as follows: for every $\theta$-nbd $U$ and a set $E\subset [a,b]$ of measure zero, there exists a gauge $\delta$ such that for every $\delta$-fine partial partition $D=\left\{\right([x_{i-1},x_i],t_i):1\le i\le n\}$ of $[a,b]$ with $t_i\in E$, there exist $\theta$-nbds $U_1,U_2,\dots ,U_n$ such that $\sum _{i=1}^n{U}_i\subseteq V$ and $F\left(x_i\right)-F\left(x_{i-1}\right)\in U_i$ for each $i=1,2,\dots ,n$. In this paper, we introduce the SL integral of a function taking values on a locally convex topological vector space (LCTVS). Further, we show that this integral is equivalent to a stronger version of the Henstock integral.