# About Henstock integrable vector-valued function

In what follows, $X$ is a Hausdorff locally convex topological vector space over the reals whose topology is generated by a family $P$ of all continuous seminorms on $X$. We consider the following definition.

Definition 1. Let $f: [a,b]\to X$ be a function. We say that $f$ is Henstock integrable on $[a,b]$ if there exists $w\in X$ such that for any $p\in P$ and $\epsilon >0$ we can find a positive function $\delta$ defined on $[a,b]$ such that for any Henstock $\delta$-fine partition $D=\{(t_i,I_i): i=1,\cdots,n\}$ of $[a,b]$, we have $$p\left(\sum_{i=1}^nf(t_i)|I_i| - w \right)<\epsilon.$$ In this case, we write $$w=(H)\int_{a}^bf.$$

Let us adopt the following notations.

• $X'$ refers to the topological dual of $X$, i.e., the space of all continuous linear functionals on $X$.
• $H([a,b],X)$ the space of all Henstock integrable functions $f:[a,b]\to X.$
• $H([a,b],\mathbb{R})$ the space of all Henstock integrable functions $g:[a,b]\to \mathbb{R}.$

A direct consequence of Definition 1 leads to the following remark.

Remark 2. If $f\in H([a,b],X)$ then for each $x'\in X'$, the composition $x'(f):[a,b]\to \mathbb{R}$ is Henstock integrable, i.e., $x'(f)\in H([a,b],\mathbb{R}).$

The following is a result that I have formulated.

Result 3. If $f\in H([a,b],X)$, then for each $p\in P$, the set $\{x'(f):x'\in U_{p}^0\}$ is uniformly Henstock integrable on $[a,b]$.

Here, $U_{p}^0$ refers to the polar of the set $$U_p=\{x\in X: p(x)\le 1\}.$$

I got no problem of proving Result 3. I tried working on the converse of Result 3 but can't do it. In some sense, does the converse of Result 3 holds? My intuition is that it holds if $X$ is a metrizable LCTVS (of which I don't how to prove the problem). Any tips are very much appreciated...

• What is the definition of uniformly Heinstock integrable family of functions? – Norbert May 15 '13 at 3:09
• Some authors use the term Henstock equi-integrability instead of uniformly Henstock integrable family of functions. This is how they defined. A family $\mathcal{F}\subseteq H([a,b],\mathbb{R})$ is said to be uniformly Henstock integrable on $[a,b]$ iff for every $\epsilon >0$ there exists a positive function $\delta$ defined on $[a,b]$ such that for every $g\in \mathcal{F}$ and every $\delta$-fine Perron partition $D=\{(t,[u,v])\}$ of $[a,b]$, we have $$\left|(D)\sum g(t)(v-u)-\int_{a}^b g\right|<\epsilon.$$ – Juniven May 15 '13 at 3:39