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\Year{2010/2011} % publication information for your paper
\Editor{Udayan B. Darji} % for your paper
\Received{November 29, 2010} % publication information for your paper
%\CoverAuthor{Caterina La Russa\\}
%\CoverTitle{Henstock Type Integral for Vector Valued Functions in a Compact Metric Space}
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\begin{Author}
% First and last name of author
\FirstName{Caterina}\LastName{La Russa}
% full postal address including postal code and country
\PostalAddress{Department of Mathematics, University of Palermo, via Archirafi 34, 90123 Palermo,
Italy}
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\Email{larussa@math.unipa.it}
\end{Author}
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\primary{28B20}
\secondary{26A39, 28B05, 46G10, 54C60}
\end{MathReviews}
% Put key words and phrases here, one keyword at a time.
\begin{KeyWords}
\keyword{Henstock integral}
\keyword{McShane integral}
\keyword{partitions}
\keyword{gauge}
\end{KeyWords}
\markboth{Caterina La Russa}{Henstock-Type integral for vector valued functions}
\title{Henstock-Type integral for vector valued
functions in a compact metric space}
\begin{document}
\maketitle
\begin{abstract}
We define a Henstock-type integral for vector valued functions
defined in a probability metric compact Radon space, using a
suitable family ${\mathcal B}$ of measurable sets which play the
role of ``\textit{intervals}''. When ${\mathcal B}$ is the family of
all subintervals of $[0,1]$ we obtain the classical
Henstock--Kurzweil integral on the real line, whereas if ${\mathcal
B}$ is the family of all subintervals of $[0,1]^2$, or that of all
subintervals of $[0,1]^2$ with a fixed regularity, we obtain the
classical Henstock integral on the plane with respect to the
Kurzweil base or the Kempisty base respectively.
\end{abstract}
\section{Introduction}
The theory of integration introduced by Lebesgue in 1902 is a
powerful tool which, perhaps because of its abstract character, does
not have the intuitive appeal of the Riemann integral. Moreover, as
Lebesgue himself observed in his thesis \cite{le}, his integral does
not integrate all unbounded derivatives and so it does not provide a
solution for the problem of recovering a function from its
derivative. Besides the Lebesgue theory does not cover nonabsolutely
convergent integrals. In 1957 Kurzweil \cite{ku} and, independently,
in 1963 Henstock \cite{he} gave a new definition of integral, which
is more general than that of Lebesgue. Its construction uses Riemann
sums associated to interval partitions which are pointwise fine
instead of uniformly fine (as in case of the classical Riemann
integral). The Henstock–-Kurzweil integral has the power of
Lebesgue's one and includes it. Moreover it integrates all
derivatives. The interval partitions used in the Riemann sums are
closely connected with the topology of the real line and in an easy
way it is possible to generalize the definition of the Henstock
integral to real (or vector) valued functions defined in $\R^{2}$
(see \cite{o}).
When the functions are defined in a more general setting different
from the euclidean one, the situation becomes more complicated. The
biggest difficulties are to find a suitable family of measurable
sets which play the role of ``\textit{intervals}'' in the
construction of the Riemann sums and to prove the existence of
partitions that are fine with respect to a fixed gauge.
In 2000 N. W. Leng and L. P. Yee, defined a Henstock-type
integral on a metric measure space (\cite{l}). When the space is
the plane, their generalized intervals are the family of polygons
with vertical or horizontal edges or the family of simply
connected domains in the plane with piecewise circular edges (see
Remark \ref{r1}).
The aim of the present paper is to provide a definition of the
classical Henstock integral in the case of vector valued
functions defined in a probability metric compact Radon space.
To this end in Section $3$ we define ``\textit{intervals}'' the sets
of a family ${\mathcal B}$ satisfying suitable properties. Then we
consider a method of integration (the ${\mathcal B}_{H}$-integral)
which involves finite Henstock partitions by non overlapping sets of
${\mathcal B}$. If ${\mathcal B}$ is the family of all subintervals
of $[0,1]$, the corresponding ${\mathcal B}_{H}$-integral is the
classical Henstock--Kurzweil integral on the real line; whereas if
${\mathcal B}$ is the family of all subintervals of $[0,1]^2$ or of
all subintervals of $[0,1]^2$ with a fixed regularity $\alpha$,
where $0<\alpha<1$, we obtain the Henstock integral on the plane
with respect to the Kurzweil base or the Kempisty base,
respectively.
\noindent Besides, using the Axiom of Choice, we construct, in any
probability metric compact Radon space, a family of sets satisfying
the properties of the ``\textit{intervals}'' (see Theorem \ref{e}).
In Section $4$, using McShane partitions of ``\textit{intervals}''
instead of Henstock partitions, we obtain a kind of McShane integral
that is equivalent to the generalized McShane integral introduced by
Fremlin in \cite[Theorem 2]{fr1}.
\section{ Preliminaries}
\noindent Throughout this paper $(\Omega,d,\Sigma,\mu)$ is a
probability metric compact Radon space i.e.: \begin{enumerate}
\item
$(\Omega,d)$ is a metric space;
\item
$(\Omega,\Sigma,\mu)$ is a probability complete space;
\item ${\mathcal T}\subset\Sigma$, where ${\mathcal T}$ is the topology induced by the metric $d$
on $X$;
\item the measure $\mu$ is regular, i.e. $$\mu(E)=\sup\{\mu(F):F\subseteq E\, \ F\mbox{
closed}\}=\inf\{\mu(G): E\subseteq G\, \ G \in {\mathcal T}\}$$ for
every $E\in\Sigma$;
\item
$\mu$ is $\tau$-{\it additive}, i.e.\ if ${\mathcal G}\subseteq
{\mathcal T}$ is non-empty and upwards directed by inclusion, then
$$
\mu(\bigcup_{G\in{\mathcal G}}G)=\sup\{\mu(G):G\in{\mathcal G}\}.
$$
\end{enumerate}
For $A \subset \Omega$, $A^0$, $\overline{A}$, $A^c$ and $\partial
A$ are the interior part, the closure, the complement and the
boundary of $A$, respectively. For $A, B \subset \Omega$, we
denote by $A\triangle B$ the symmetric difference of $A$ and $B$.
\\ If $A \subset \Omega$, $diam(A):=\sup \{ d(x,y): x,y \in A \}$
is the \textit{diameter} of $A$. Each set $B_{r}(w)= \{z \in
\Omega : d(w,z)0$, is called
an open ball. Denote by ${\mathcal T_{1}}$ the family of all open
balls. Throughout this paper, moreover we assume that the measure
$\mu$ satisfies the following property: \begin{equation*}\label{a}
(A) \ \ \ \ \ \ \mu(S)>0 \ {\rm and} \ \mu(S)= \mu(\overline{S}) \
{\rm for \ all} \ S \in {\mathcal T_{1}}.
\end{equation*}
\noindent Denote by $\Sigma_{\partial}$ the family of all $A \in
\Sigma$ such that $\mu(A)>0$ and $\mu(\partial A)=0$. \\ We say
that two measurable sets $A$ and $B$ are \textit{non overlapping}
if $A^0 \cap B^0 = \emptyset $ and $\mu(\partial A \cap \partial
B)=0$.
\vspace{1ex} Given a non-empty family ${\mathcal A} \subseteq
\Sigma$, a finite collection ${\mathcal P}=\{(A_n, \omega_n):
n=1,\ldots,p \}$ of pairwise non overlapping sets $A_n \in {\mathcal
A}$ and points $\omega_n\in \overline{A_n}$ is said to be a {\it
Henstock ${\mathcal A}$-partition} (briefly {\it ${\mathcal
A}_{H}$-partition}). If we assume only that $\omega_n\in\Omega$, for
$1\leq n\leq p$, then ${\mathcal P}$ is said to be a {\it McShane}
${\mathcal A}$-{\it partition} (briefly {\it ${\mathcal
A}_{Mc}$-partition}). If $A \in{\mathcal A}$ and $\mu(A \triangle
(\bigcup_{n=1}^p A_n))=0$, we say that ${\mathcal P}$ is an {\it
${\mathcal A}_{H}$-partition} (resp. {\it ${\mathcal
A}_{Mc}$-partition}) {\it of $A$}.
\noindent Denote by ${\mathcal A}^U$ the family of all finite
unions of non overlapping sets of ${\mathcal A}$.
\vspace{1ex} \noindent Each function
$\Delta:\Omega\rightarrow{\mathcal T}$ such that
$\omega\in\Delta(\omega)$ for each $\omega\in\Omega$ is called
{\it gauge}.
\noindent Let $\Delta$ be a gauge and let $E \subset \Omega$. An
${\mathcal A_{H}}$-partition (${\mathcal A_{Mc}}$-partition)\\
${\mathcal P}=\{(A_n, \omega_n): n=1,\ldots,p\}$ is said to be:
\begin{description}
\item{(a)} $\Delta$-{\it fine}, if $A_n\subset \Delta(\omega_n)$ for each $1\leq n\leq
p$;
\item{(b)} {\it tagged in} $E$, if $\omega_n\in E$ for each $1\leq n\leq p $.
\end{description}
For simplicity we write $({\mathcal A}_{H}, \Delta)$-{\it
partition} (\ resp. $( {\mathcal A}_{Mc},\Delta)$-{\it partition}),
for an ${\mathcal A}_{H}$-partition (resp. ${\mathcal
A}_{Mc}$-partition) that is $\Delta$-fine.
\vspace{1ex} Given $f: \Omega\rightarrow Y$, where $Y$ is any
Banach space, and a partition ${\mathcal P}=\{(A_n,
\omega_n):n=1,\ldots,p\}$, we set $\sigma(f,{\mathcal
P}):=\sum_{n=1}^pf(\omega_n)\mu(A_n).$
\section{The family ${\mathcal B}$ of ``intervals''}
One of the most important problems in a theory of gauge integrals
is the existence of partitions, fine with respect to a fixed
gauge. So in our framework, first of all, it is essential to
define a suitable family of measurable sets which play the role of
``\textit{intervals}'' in the construction of Riemann sums.
\begin{deff}{\rm We say that ${\mathcal B}
\subset \Sigma$ is a family of \textit{intervals} in
$(\Omega,d,\Sigma,\mu)$ if it satisfies the following properties:
\begin{description}
\item{(j)} ${\mathcal B} \subseteq \Sigma_{\partial}$;
\item{(jj)} $\Omega \in {\mathcal B}$;
\item{(jjj)} for each $B \in {\mathcal B}$, there exist in ${\mathcal B}$
non overlapping subsets $B_1,...,B_k$ of $\overline{B}$ such that
\begin{equation} \label{d} \mu(B \setminus \bigcup_{i=1}^k B_i)=0
\ \ \ and \ \ \ diam(B)>c\cdot diam(B_i)\end{equation} for every
$i=1,...,k$, where $c>1$ is a fixed constant;
\item{(jv)} if $B \in {\mathcal B}$, for each $C\subseteq B$ and
$C \in {\mathcal B}^{U}$, then $B\setminus C$ belongs to
${\mathcal B}^U$ unless a set of zero measure.
\end{description}}\end{deff}
\begin{ex} {\rm Let $\Omega=[0,1]$ be endowed with the
Lebesgue measure and the Euclidean topology and let ${\mathcal B}$
be the family of all subintervals of $\Omega$. Then ${\mathcal B}$
satisfies properties $(j)$--$(jv)$.} \end{ex}
\begin{ex} {\rm Let $\Omega=[0,1]^{2}$ be endowed with the
Lebesgue measure and the Euclidean topology and let ${\mathcal B}$
be the Kurzweil base (i.e.\ the family of all subintervals of
$\Omega$). Then ${\mathcal B}$ satisfies properties
$(j)$--$(jv)$.}
\end{ex}
\begin{ex} {\rm Let $\Omega=[0,1]^{2}$ be endowed with the
Lebesgue measure and the Euclidean topology and let ${\mathcal B}$
be the Kempisty base (i.e.\ the family of all the subintervals of
$[0,1]^2$ whose regularity\footnote{Recall that if
$I=[a_{1},b_{1}]\times[a_{2},b_{2}]\subset \R^{2}$ the {\it
regularity} of $I$ is the number $r(I)=\min_{i=1,2}(b_{i}-a_{i})/
\max_{i=1,2}(b_{i}-a_{i})$.} is greater than a fixed $\alpha$, for
$0<\alpha<1$). Then ${\mathcal B}$ satisfies properties
$(j)$--$(jv)$.} \end{ex}
\vspace{2ex} In \cite{l}, N. W. Leng and L. P. Yee consider a
metric measure space $(Y,d,\mu)$ that satisfies condition (A) and
such that:
\vspace{1ex} (\textit{B}) \ \ for every measurable set $W \subset
Y$ and every $\epsilon>0 $, there exist an open set $U$ and a
closed set $Z$ such that $Z \subset W \subset U$ and
$\mu(U\setminus Z)<\epsilon$.
\vspace{2ex} Note that each probability, metric, compact Radon
space $(\Omega,d,\Sigma,\mu)$, satisfies condition (\textit{B}).
\vspace{2ex}\noindent In their framework, N. W. Leng and L. P. Yee
define the following families of sets: $$\mathcal{H}_{1}= \left
\{\overline{B_{1}}\setminus \overline{B_{2}}: B_{1} , B_{2} \in
{\mathcal T}_{1} {\rm \ \ and} \ \ B_{1} \nsubseteq B_{2}, \ B_{2}
\nsubseteq B_{1} \right \},$$
$$\mathcal{H}_{2}= \left \{ \bigcap_{i\in \Lambda} X_{i}\neq
\emptyset: X_{i} \in \mathcal{H}_{1} \ {\rm and} \ \Lambda {\rm
\ is \ a \ finite \ index \ set} \right \}.$$
\noindent They call the members of $\mathcal{H}_{2}$
\textit{generalized intervals} and a finite union of mutually
disjoint generalized intervals \textit{elementary set}. Then using
the elementary sets, they are able to define a Henstock-type
integral on $Y$.
\begin{rem}\label{r1} {\rm If $Y= \R^{2}$ with the metric $d_{1}(x,y)= \max \{ |x_{1}-y_{1}| ,
|x_{2}-y_{2}| \}$, a generalized interval of $\mathcal{H}_{2}$
looks like a polygon with vertical or horizontal edges, and each
edge is not necessarily included. Instead, if we consider $Y=
\R^{2}$ with the metric $d_{2}(x,y)= [(x_{1}-y_{1})^{2} +
(x_{2}-y_{2})^{2}]^{\frac{1}{2}}$, a generalized interval is a
simply connected domain in the plane with piecewise circular edges
and each edge is not necessarily included. In any case we obtain a
base, different from both the Kurzweil and the Kempisty
base.}\end{rem}
\noindent In \cite{l} the following result is shown:
\begin{lem} \label{l1} {\rm (\cite{l}, Theorem p. $36$)} Given a gauge $\Delta$ on a
generalized interval $I \in \mathcal{H}_{2}$, there exists a
$\Delta$-fine division\footnote{A \textit{division} of $I$ is a
finite collection $\{(I_n, x_n): n=1,\ldots,p \}$ of pairwise
mutually disjoint subintervals $I_n \subset I$ and points $x_n \in
I_n$ such that $\bigcup_{n=1}^{p} I_{n} =I$.} of $I$.\end{lem}
\begin{lem} \label{2} Given a gauge
$\Delta$ on $(\Omega,d,\Sigma,\mu)$, there exists a $\Delta$-fine
division of $\Omega$.\end{lem}
\Pf \ Taking in account that $\Omega$ is compact, the proof
follows as in Theorem on page 36 of \cite{l}, after suitable
changes. \hfill$\Box$
\begin{lem}\label{H} In the space $(\Omega,d,\Sigma,\mu)$ the family
$\mathcal{H}_{2}\bigcup \Omega$ satisfies the properties
$(j)$--$(jjj)$.\end{lem}
\Pf \ Properties (j) and (jj) hold. In order to show that also the
property (jjj) is satisfied, fix $B \in \mathcal{H}_{2}\bigcup
\Omega$ and a constant $c>1$. It is enough to consider a gauge
$\Delta$ such that $\Delta(w):= B_{r}(w)$ for each $w \in B$, with
$r=\frac{diam(B)}{4c}$ and to apply Lemma \ref{l1} or Lemma
\ref{2}. \hfill$\Box$
\vspace{1ex} Now, using the Axiom of Choice, we are going to
construct, in any probability metric compact Radon space, a family
${\mathcal B}$ of intervals.
\begin{prp}\label{i} Let ${\mathcal A}$ be a family of subsets of $\Omega$
satisfying properties $(j)$--$(jjj)$. Then there exists a
subfamily ${\mathcal B} \subseteq \mathcal{A}$ which satisfies
also the property (jv).\end{prp}
\Pf \ We construct the subfamily ${\mathcal B}$ by induction.
\noindent \textit{First step}: Let $B_{1}^{k_1}=\Omega$, where
$k_1=1$.
\noindent \textit{Second step}: Since $\Omega \in {\mathcal A}$,
by the property (jjj), (and the Axiom of Choice) we can choose in
${\mathcal A}$, non overlapping subsets of $\Omega$,
$B^{1}_2,...,B^{k_2}_2$ such that $\mu(\Omega \setminus
\bigcup_{m=1}^{k_2} B^{m}_2)=0$ and $ diam(\Omega)>c\cdot
diam(B^{m}_2) $, for every $m=1,...,k_{2}$, and where $c$ is a
fixed constant greater then $1$.
\noindent \textit{Third step}: For each $B^{m}_2$, with $m=1,2,...
k_{2}$, by the property (jjj) we can choose in $\mathcal A$ a finite
number of sets satisfying the condition (\ref{d}) of property (jjj),
with $B$ replaced by $B^{m}_2$ with $m=1,2,... k_{2}$. Let call by
$B^{1}_3,...,B^{k_3}_3 \in {\mathcal A}$ all the sets obtained in
this step.
\noindent Then call by $B^{1}_n,...,B^{k_n}_n \in {\mathcal A}$,
with $k_n \in \N$, the sets obtained in the nth step.
\noindent The subfamily ${\mathcal B} = \left\{B^{m}_n
\right\}_{m,n} \subset \mathcal A$ with $n \in \N$ and $m= 1, 2,...,
k_{n}$ satisfies also the property (jv).\hfill $\Box$
\begin{thm}\label{e} In any space $(\Omega,d,\Sigma,\mu)$, there exists a family of
intervals.\end{thm}
\Pf \ It follows by Proposition \ref{i} and by Lemma \ref{H}.
\hfill$\Box$
\begin{prp}\label{p1}
Let ${\mathcal B}$ be a family of intervals in
$(\Omega,d,\Sigma,\mu)$. For each $B \in {\mathcal B}$ and for
each gauge $\Delta$ there exists a $({\mathcal
B}_{H},\Delta)$-partition of $B$.\end{prp}
\Pf \ Let $B \in {\mathcal B}$ and assume by contradiction that
there is not such a partition of $B$. By property $(jjj)$ there
exists in ${\mathcal B}$ a set $B^{(1)} \subset \overline{B}$ with
$diam(B)>c \cdot diam(B^{(1)})$ and such that there is not a
$({\mathcal B}_{H},\Delta)$-partition of $B^{(1)}$. Proceeding by
induction, we can construct a sequence $\{B^{(k)}\}_{k }$ of
${\mathcal B}$ sets such that $\overline{B^{(k)}} \supset
\overline{B^{(k+1)}}$, $diam(B^{(k)})>c\cdot diam(B^{(k+1)}) $ and
there is not a $({\mathcal B}_{H},\Delta)$-partition of $B^{(k)}$
for each $k \in \N$.
As $\Omega$ is compact, $\bigcap_{k=1}^{\infty}
\overline{B^{(k)}}\neq \emptyset $. Let $\omega_0 \in
\bigcap_{k=1}^{\infty} \overline{B^{(k)}}$. By construction
$\lim_{k\rightarrow \infty} diam(B^{(k)})=0$. So there exists an
index $k_0$ such that $B^{(k_0)} \subset \Delta(\omega_0)$ and the
pair $(B^{(k_0)}, \omega_0)$ is a $({\mathcal
B}_{H},\Delta)$-partition of $B^{(k_0)}$ and this is a
contradiction.\hfill $\Box$
\begin{prp}\label{p2}Let ${\mathcal B}$ a family of intervals in $(\Omega,d,\Sigma,\mu)$. Let $\Delta$
be a gauge and let ${\mathcal P}$ be a $({\mathcal B}_{H},
\Delta)$-partition of $B \in {\mathcal B}$. Then the partition
${\mathcal P}$ can be extended to a $({\mathcal
B}_{H},\Delta)$-partition of $\Omega$.\end{prp}
\Pf \ It follows at once from property $(jv)$ and Proposition
\ref{p1}. \hfill $\Box$
\begin{deff} {\rm A family ${\mathcal F \subset \Sigma}$ is said to be \textit{weakly fine} on $\Omega$ if for each $w \in
\Omega$ and for each $G \in {\mathcal T} $ with $w \in G$, there
exists $F \in {\mathcal F}$ such that $\mu(F)>0$, $\omega\in
\overline{F}$ and $F\subseteq G$.}\end{deff}
\begin{prp}\label{wf} Let ${\mathcal B}$ be a family of intervals in $(\Omega,d,\Sigma,\mu)$. Then
${\mathcal B}$ is weakly fine on $\Omega$.\end{prp}
\Pf \ Consider $w \in \Omega$ and $G \in {\mathcal T} $ with $w
\in G$. By properties (jj) and (jjj) there exist in ${\mathcal B}$
non overlapping sets $B_1,...,B_k$ such that $\mu(\Omega \setminus
\bigcup_{i=1}^k B_i)=0$ and $diam(\Omega)>c\cdot diam(B_i) $, for
every $i=1,...,k$ and $c>1$ fixed constant. Note that by condition
(A), each non empty set of $\mathcal{T}$ has positive measure.
Then $\mu(\Omega \setminus \overline{\bigcup_{i=1}^k B_i})=0$
implies that $\Omega \setminus \overline{\bigcup_{i=1}^k B_i}=
\emptyset$. So $w \in \overline{B_i}$ for some $i=1,...,k$.
Call $I^{(1)}$ one of such intervals. Then $w \in
\overline{I^{(1)}}$ and $diam(\Omega)> c \cdot diam(I^{(1)})$. By
property (jjj) there exist in ${\mathcal B}$ non overlapping subsets
$B^{1}_1,...,B^{k_{1}}_1$ of $\overline{I^{(1)}}$ such that
$\mu(I^{(1)} \setminus \bigcup_{j=1}^{k_{1}} B^{j}_1)=0$ and
$diam(I^{(1)})> c \cdot diam(B^{j}_1) $, for every $j=1,...,k_{1}$.
\noindent Call $I^{(2)}$ one of such intervals. Then $w \in
\overline{ I^{(2)} }$ and $diam(I^{(1)})> c \cdot diam(I^{(2)})$.
Now again by property (jjj) there exist in ${\mathcal B}$ non
overlapping subsets $B^{1}_2,...,B^{k_{2}}_2$ of $\overline{
I^{(2)} }$ such that $\mu \left( I^{(2)} \setminus
\bigcup_{n=1}^{k_{2}} B^{n}_2 \right)=0$ and $diam( I^{(2)})> c
\cdot diam(B^{n}_2) $, for every $n=1,...,k_{2}$.
\noindent Call $I^{(3)}$ one of such intervals. Then $w \in
\overline{I^{(3)}}$ and $diam(I^{(2)})> c \cdot diam(I^{(3)})$.
\noindent Proceeding by induction, we construct a sequence
$\{I^{(m)}\}_{m \in \N}$ of ${\mathcal B}$ sets with decreasing
diameter such that $\overline{I^{(m)}} \supset \overline{I^{(m+1)}}$
and $w \in \overline{I^{(m)}}$ for all $m \in \N$. Then taking in
account that $G$ is open and $w \in G$, there exists an index
$m_{0}$ such that $\omega\in \overline{I^{(m_{0})}}$ and
$I^{(m_{0})} \subset G$. \hfill $\Box$
\vspace{2ex} \noindent In the following we will use the property:
\begin{deff} {\rm A family ${\mathcal F}\subseteq\Sigma$ {\it separates
points off closed sets} if given $\varepsilon>0$,
$\omega\in\Omega$ and an open set $O$ of positive measure and
containing $\omega$, there exists $F\in{\mathcal F}$ such that
$\omega\in \overline{F}$, $F\subseteq O$ and $\mu(O\setminus
F)<\varepsilon\,.$ }\end{deff}
Note that the previous definition is slightly weaker then the
definition given in \cite{bdm}. In fact there is also the
condition that $\omega\in \overline{F}$.
\begin{prp}\label{p3} Let ${\mathcal B}$ be a family of intervals in $(\Omega,d,\Sigma,\mu)$. Then the
family ${\mathcal B}^U$ separates points off closed sets.\end{prp}
\Pf \ Let $\omega_0 \in \Omega$ and $O \in {\mathcal T}$ be
given, with $\omega_0 \in O$, and let $\varepsilon>0$ be fixed.
Since $\mu$ is regular, let $F$ be a compact $F \subset O$ such
that $\mu(O \setminus F)< \varepsilon$. Now define a gauge
$\Delta$ in $\Omega$ in the following way: $\Delta(\omega) \subset
O$ if $\omega \in O$, $\Delta(\omega) \subset \Omega \setminus F$,
if $\omega \in O^c$. Since ${\mathcal B}$ is weakly fine on
$\Omega$ (Proposition \ref{wf}), then let $B_0 \subset
\Delta(\omega_0)$ be a ${\mathcal B}$ set such that $\omega_0 \in
\overline{B_0}$. Applying Proposition \ref{p2} to the partition
${\mathcal P}=\{(B_0,\omega_0)\}$ and to the gauge $\Delta$,
${\mathcal P}$ can be extended to a $({\mathcal
B}_{H},\Delta)$-partition ${\mathcal Q}$ of $\Omega$. Set $U=
\bigcup B$ where the union is extended to all the sets $B\in
{\mathcal B}$ such that $(B,\omega) \in {\mathcal Q}$ and $B
\subset O$. Therefore $U$ is the required set.\hfill $\Box$
\section{ Henstock and McShane ${\mathcal
B}$-integrals}
\vspace{2ex} \noindent Let ${\mathcal B}$ be a fixed family of
intervals in $(\Omega,d,\Sigma,\mu)$ and let $(X, \| \cdot \|)$ be
a Banach space.
\begin{deff} {\rm We say that a function $f\colon\Omega\rightarrow
X$ is} ${\mathcal B}_{H}$-integrable (${\mathcal B}_{Mc}$-
integrable) on $\Omega$ {\rm if there exists $w \in X$ satisfying
the following property:
\noindent for each $\varepsilon>0$ there exists a gauge
$\Delta\colon\Omega\rightarrow{\mathcal T}$ such that
%\marginpar{b}
\begin{equation}\label{b}
\| \sigma(f,{\mathcal P})-w\|<\varepsilon,
\end{equation}
for every $({\mathcal B}_{H}, \Delta)$-partition ($({\mathcal
B}_{Mc},\Delta)$-partition) ${\mathcal P}$ of $\Omega$. We set
$$w=({\mathcal B}_{H})\int_{\Omega}f\, d\mu \ \ \ \ \ \ \ \ \left( w=({\mathcal
B}_{Mc})\int_{\Omega}f\, d\mu \right).$$
\noindent Given a measurable set $E\subset \Omega$ we say that
$f$ is ${\mathcal B}_{H}$-integrable (${\mathcal
B}_{Mc}$-integrable) on $E$ if the function $f\chi_E$ is
${\mathcal B}_{H}$-integrable (${\mathcal B}_{Mc}$-integrable) on
$\Omega$, where as usual, $\chi_E$ is the characteristic function
of the set $E$. \\ We set $w(E)=({\mathcal
B}_{H})\int_{\Omega}f\chi_E\,d\mu$ \ $\left( w(E)=({\mathcal
B}_{Mc})\int_{\Omega}f\chi_E\,d\mu \right)$. }\end{deff}
\begin{rem} \label{w2} {\rm If $\Omega=[0,1]$ is endowed with the
Lebesgue measure and the Euclidean topology, $X=\R$ and
${\mathcal B}$ is the family of all subintervals of $\Omega$, then
the ${\mathcal B}_{H}$-integral is the classical
Henstock--Kurzweil integral on the real line. If
$\Omega=[0,1]^{2}$ is endowed with the Lebesgue measure and the
Euclidean topology and ${\mathcal B}$ is the Kurzweil base or the
Kempisty base, then the ${\mathcal B}_{H}$-integral is the
Henstock integral on the plane with respect to the Kurzweil base
or the Kempisty base (see \cite{o}).}\end{rem}
\begin{prp}\label{p4} A function $f\colon\Omega\rightarrow X$ is ${\mathcal
B}_{H}$-integrable on $B \in {\mathcal B}$ if and only if the
following Cauchy condition holds:
\noindent for each $\varepsilon>0$ there exist a gauge $\Delta$
such that
%\marginpar{j}
\begin{equation}\label{j}
\left\| \sigma(f,{\mathcal P})-\sigma(f,{\mathcal Q})\right\|<
\varepsilon,\end{equation} for each couple ${\mathcal P}$,
${\mathcal Q}$ of $({\mathcal B}_{H}, \Delta)$-partitions of $B$.
\end{prp}
\noindent \Pf \ The proof follows as in Proposition 2 of
\cite{bdm} after suitable changes. \hfill$\Box$
\vspace{2ex} \noindent The above property guarantees that
\begin{prp}\label{p5}
Let $f\colon\Omega\rightarrow X$ be a ${\mathcal B}_{H}$-integrable
function on $\Omega$. Then the function $f \chi_B$ is ${\mathcal
B}_{H}$-integrable on $\Omega$ for every set $B \in {\mathcal
B}$.\end{prp}
\Pf \ The proof follows from Proposition \ref{p2} and Proposition
\ref{p4}.\hfill$\Box$
\vspace{2ex} Note that the ${\mathcal B}_{H}$-integral is uniquely
determined, closed under addition and scalar multiplication.
Moreover also the Henstock Lemma version for vector valued function
holds (see \cite{s}).
\vspace{2ex} In \cite{fr1} D.H. Fremlin studies, in a
$\sigma$-finite outer regular quasi-Radon space, a method of
integration for vector-valued functions which is a generalization of
the McShane process of integration \cite{mc}. This method involves
infinite McShane partitions by disjoint families of measurable sets
of finite measure. However, in the compact case, the method may use
finite McShane partitions with disjoint measurable sets (see
\cite[Proposition E1]{fr1}).
\begin{deff} {\rm We say that a function $f\colon\Omega\rightarrow X$ is
{\it Fremlin-integrable} on $\Omega$ (\cite[Proposition E1]{fr1}) if
there exists $w\in X$ satisfying the following property:
\noindent for each $\varepsilon>0$ there exists a gauge $\Delta$
such that
%\marginpar{c}
$$\left\| \sigma(f,{\mathcal P})-w\right\|<\varepsilon,$$
for every finite $(\Sigma_{Mc}, \Delta)$-partition $\mathcal P$ of
$\Omega$.}
\end{deff}
Now we compare the ${\mathcal B}_{Mc}$-integral with the
Fremlin-integral. We need the following Lemma that may be proved
in a standard way (for the case $\Omega=[0,1]$ and $X=I\!\!R$ see
\cite{g}, p. 323).
\begin{lem}\label{l2} Let $f\colon \Omega \to X $ be a function and let $N \subset
\Omega$. If $\mu(N)=0$, then for each $\varepsilon
>0$ there exists a gauge $\Delta$ in $N$ such that $ \sigma(\|f\|
, {\mathcal P})< \varepsilon$, for each $({\mathcal B}_{Mc},
\Delta)$-partition ${\mathcal P}$ tagged in $N$.\end{lem}
\begin{thm}\label{t1} A function $f\colon\Omega\rightarrow X$ is ${\mathcal
B}_{Mc}$-integrable on $\Omega$ if and only if it is
Fremlin-integrable on $\Omega$.
\end{thm}
\Pf \ Let $f$ be Fremlin-integrable on $\Omega$. Since for any
gauge $\Delta$, each $({\mathcal B}_{Mc}, \Delta)$-partition is
also a $(\Sigma_{Mc}, \Delta)$-partition, therefore $f$ is
${\mathcal B}_{Mc}$-integrable on $\Omega$.
For the converse, let $\varepsilon>0$ be fixed and let $\Delta$
be a gauge such that
\begin{equation}\label{1}
\left\|\sigma(f,{\mathcal P})-({\mathcal
B}_{Mc})\int_{\Omega}f\right\|<\frac{\varepsilon }{4} \ ,
\end{equation}
\noindent for each $({\mathcal B}_{Mc},\Delta)$- partition
${\mathcal P}$ of $\Omega$.
\noindent Now let ${\mathcal Q}=\{(E_i,\omega_i): \ i=1,...,n\}$ be
a $(\Sigma_{Mc},\Delta)$-partition of $\Omega$. Put $m=
\max_{i=1,...,n} \|f(\omega_i)\|$ and take $0< \eta < (4n m)^{-1}
\varepsilon $.
\noindent The proof will be inductive.
Assume that for some $1\leq q