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''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|Denjoy theorem on derivatives]] of a function of one variable. The theorems were established by A.J. Ward .

====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  A.J. Ward,   "On the differentiation of additive functions of rectangles"  ''Fund. Math.'' , '''28'''  (1936)  pp. 167–182</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  A.J. Ward,   "On the derivation of additive functions of intervals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097090/w09709018.png" />-dimensional space"  ''Fund. Math.'' , '''28'''  (1937)  pp. 265–279</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Saks,   "Theory of the integral" , Hafner  (1952)  (Translated from French)</TD></TR></table>