Evidently a packing is either finite or countably infinite. Using the lan-
guage of Henstock [7], we call a finite packing 7r a division of an interval
[a, b] if cr(7r) = [a, b]. Some authors call this a partition; throughout we shall
employ this term in its usual set-theoretic sense.
2,2 Full and fine covering relations
Most of the covering relations that one encounters in analysis have certain
local properties. The covering relations and covering lemmas that are our
central concern are those that arise naturally in discussions of limits or deriva-
tives. We now introduce the notion of a full covering relation and its dual
notion, that of a fine covering relation. They will play a central role in all of
our studies. Since our focus is on limits of the form
lim I/W-/WI,
y,z—KT, yxz (y _
we require the following special form of the covering relations. Note in par-
ticular that the pairs (I,x) in the relations always have x an interior point
of the closed interval / .
2.4 DEFINITION. A covering relation (3 is said to be full at a point x
provided that there exists a S 0 so that ([y, z\ x) £ (3 for every y x z
with 0 z y 6 . Such a relation is said to be a full covering relation on
a set E if it is full at each point of E.
We give also a version parallel to this; this is essentially the notion of a
Vitali covering.
2.5 DEFINITION. A covering relation 0 is said to be fine at a point x
provided that for every e 0 there is a pair ([j/,2r],x) G /?, with y x z
and 0 z y e. Again (3 is a fine covering relation on a set E if it is fine
at each point of E.
The relation between full and fine covering relations can be expressed by
the following simple observation. Let 0 be a covering relation and let
= (J x R) \
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