Ward theorem

on the differentiation of an additive interval function

Let $ F $ be a real-valued additive interval function, and let $ {\underline{F} } ( x) $( $ {\overline{F}\; } ( x) $) be the greatest lower (least upper) bound of the limits of the sequences $ F( G _ {n} ) /| G _ {n} | $, where $ | G _ {n} | $ is the Lebesgue measure of $ G _ {n} $, and $ \{ G _ {n} \} $ runs through all regular sequences of intervals contracting towards the point $ x $. Then the equation $ \overline{F}\; ( x) = \underline{F} ( x) $ is valid almost-everywhere (in the sense of the Lebesgue measure) on the set $ \{ {x } : {\underline{F} ( x) > - \infty \textrm{ or } \overline{F}\; ( x) < \infty } \} $. A sequence of intervals $ G _ {n} $ is regular if there exist a number $ \alpha > 0 $ and sequences of spheres $ S _ {n} ^ \prime $, $ S _ {n} ^ {\prime\prime} $ such that for all $ n $,

$$ \mathop{\rm diam} S _ {n} ^ \prime > \alpha \mathop{\rm diam} S _ {n} ^ {\prime\prime} $$

and

$$ S _ {n} ^ \prime \subset G _ {n} \subset S _ {n} ^ {\prime\prime} . $$

If, in the above formulation, the condition of regularity is discarded, Ward's second theorem is obtained. These theorems generalize the Denjoy theorem on derivatives of a function of one variable. The theorems were established by A.J. Ward .

References

[1a]	A.J. Ward, "On the differentiation of additive functions of rectangles" Fund. Math. , 28 (1936) pp. 167–182
[1b]	A.J. Ward, "On the derivation of additive functions of intervals in -dimensional space" Fund. Math. , 28 (1937) pp. 265–279

References

[a1]	S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)