DERIVATES OF INTERVAL FUNCTIONS 5

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.