DERIVATES OF INTERVAL FUNCTIONS

7

Then 7 is full at a point x if and only if j3 is not fine at x. This may be thought

of as a duality (or more properly a negative duality) and can be made the

basis for a genuine dual relationship between pairs of differentiation bases (as

in [21] and [22]); we shall not need any such formal apparatus. Nonetheless

most statements for upper and lower derivates will appear in dual pairs that

correspond to the notions of full and fine covering relations and measures

will appear in dual pairs that are similarly related.

By a covering lemma we mean an assertion that from one or several

covering relations some subcovering may be constructed. The model is the

classical Vitali covering theorem. Expressed in our language this theorem

asserts that if P is a fine covering relation on the set E then there is a

collection

7 = {(Ii,Xi) : i = l , 2 , . . . n }

contained in p for which the intervals U are pairwise disjoint and the sum

J2(i,x)ey \I\ approximates the measure of E arbitrarily closely. We wish the

phrase to apply rather broadly. In the spirit of this language then each of

the lemmas below is a covering lemma. As we shall need to refer to these

later on we present them as a series of lemmas. The proofs are elementary

but, to familiarize the reader with the language, are included.

L E M M A 2.6 Let fli and p2 be full covering relations on a set E. Then

Pi H P2 is a full covering relation on E.

Proof. If /3i and P2 are full covering relations on a set E then there are,

for every x G £", positive numbers 6i(x) and 82(x) so that ([y,z],x) G Pi for

every y x z with 0 z — y $i(x) and so that ([y, z],x) G P2 for every

y x z with 0 z — y 62(x) . Evidently ([y, z],x) G pi D p2 for every

y x z with 0 z — y min{^i(a:), S2(x)} and so Pi 0 P2 is full at each

x G E.

L E M M A 2.7 Let Pi be a full covering relation on a set E and let p2 be a

fine covering relation on E. Then Pi fl P2 is a fine covering relation on E.

Proof. Let us show that Pi fl P2 is fine at any point x G E. Since pi

is a full covering relation on E then there is a positive number 81 so that

([?/, z],x) G Pi for every y x z with 0 z — y 81. Now let e 0 and set

ti = min{5i, e}. Since p2 is fine at x there is at least one pair ([y, z],x) G P2