technique can be used to give a transparent proof of the fact, just mentioned,
that JE \f(t)\ dt recovers the variation of F on a measurable set E. More im-
portantly it permits generalizations to situations where the Lebesgue integral
is not directly applicable.
Thus the separate notions of differentiation, measure and integration are
all linked by the notion of a covering relation. Here we continue this theme
and organize it in a language that allows the generalizations to develop.
2,1 Basic language of covering relations
We introduce the general notion of a covering relation and develop the lan-
guage needed for our discussion of these relations.
2.1 DEFINITION. A covering relation on a set of real numbers £ is a
collection of pairs (7, x) where I is a closed interval and x G IDE.
If the collection of all closed intervals is called I then a covering relation
is merely a subset (3 of the product X x IR. We prefer lower case greek letters
for covering relations, usually employing a, /?, 7 or x.
2.2 DEFINITION. If /? is a covering relation and E is a set of real numbers
then /3(E), /?[£], and cr(/3) denote the following sets:
1. /3(E) = {(I,x)e/3: ICE},
2. 0[E] = { ( / , * ) / ? : x eE} .
The expressions /3(E) and f3[E] are also covering relations and are subsets
of /?. The passage to (3(E) and (3[E] from (3 is a common device in derivation
theory. In some settings (eg. in [6, p. 12]) /3(E) is called a "pruning" of (3;
the language is meant to indicate that some inessential members of (3 have
been removed. The most common pruning of a relation (3 on a set E will be
to form /3(G) for an open set G that contains E.
2.3 DEFINITION. A packing is a covering relation 7r with the property that
for distinct pairs (Ii,x\) and (I2,x2) belonging to 7r the intervals ii and I2
do not overlap.
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