may want to check this fact as described, for instance, by the group N. Bourbaki in [32, Corollaire I, p. II.6].
1.1.1 Basic Properties
Let
be a Banach space with norm
. Here, we describe regulated functions
, where
, with
, is a compact interval of the real line
.
Definition 1.1: A function
is called
regulated, if the lateral limits
exist. The space of all regulated functions
will be denoted by
.
We denote the subspace of all continuous functions
by
and, by
, we mean the subspace of regulated functions
which are left‐continuous on
. Then, the following inclusions clearly hold
Remark 1.2: Let
. By
, we mean the set of all elements
for which
for every
. Thus, it is clear that
and the range of
belongs to
. Note that, for a given
,
and
do not necessarily belong to
.
Any finite set
of points in the closed interval
such that
is called a division of
. We write simply
. Given a division
of
, its elements are usually denoted by
, where, from now on,
denotes the number of intervals in which
is divided through the division
. The set of all divisions
of
is denoted by
.
Definition 1.3: A function
is called a
step function, if there is a division
