4
BRIAN S. THOMSON
develop a countable, but transfinite, sequence of extensions of the Lebesgue
integral which solve this problem.
A simpler analysis is available directly from the meaning of the assertion
Ff(x)
= f(x). For any positive number e the collection
0=U{y,z),x)
*M--*M_/(X)
y
e (4)
has an obvious local property at each point. Specifically at each point x and
for every sufficiently small interval / with x I the pair (I,x) belongs to /?.
One can view this as a relation connecting each point with many intervals.
For the problem at hand one can evidently recover the increment F(b)
F(a) within an error of e(b a) by forming the sum
n
^ / ( 6 ) ( ^ - ^ - i )
1 = 1
taken over a partition a = x0 x\ .. . xn = b for which each
([£,•_!,£,•],&) /3. This is the basis for the theory of generalized Riemann in-
tegrals developed by Henstock [7] and Kurzweil [11], and the argument of the
previous sentence is almost completely a proof of the fundamental theorem
of the calculus in this setting.
Subtle properties of covering relations may be explored geometrically and
applied in a unified manner to give insights into the structure of real func-
tions. The idea in essence is due to Vitali although the language is developed
here somewhat differently. The original application of Vitali arises from the
observation that from certain covering relations (3 that might hold for a set
E of real numbers one may extract a sequence
{(IUXi),(I2,X2),. . . ,(In,Xn)}
where the intervals {/,-} are nonoverlapping and provide an approximation
to the outer Lebesgue measure \(E) of the set E:
CO
A(£)«£|/,|.
1 = 1
Evidently this approximation can be used to connect measure properties of
certain sets with differentiation properties of functions. In particular such a
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