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''on the differentiation of an additive interval function''
 
''on the differentiation of an additive interval function''
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097090/w0970901.png" /> be a real-valued additive interval function, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097090/w0970902.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097090/w0970903.png" />) be the greatest lower (least upper) bound of the limits of the sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097090/w0970904.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097090/w0970905.png" /> is the Lebesgue measure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097090/w0970906.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097090/w0970907.png" /> runs through all regular sequences of intervals contracting towards the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097090/w0970908.png" />. Then the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097090/w0970909.png" /> is valid almost-everywhere (in the sense of the Lebesgue measure) on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097090/w09709010.png" />. A sequence of intervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097090/w09709011.png" /> is regular if there exist a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097090/w09709012.png" /> and sequences of spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097090/w09709013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097090/w09709014.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097090/w09709015.png" />,
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Let $  F $
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be a real-valued additive interval function, and let $  {\underline{F} } ( x) $(
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$  {\overline{F}\; } ( x) $)  
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be the greatest lower (least upper) bound of the limits of the sequences $  F( G _ {n} ) /| G _ {n} | $,  
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where $  | G _ {n} | $
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is the Lebesgue measure of $  G _ {n} $,  
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and $  \{ G _ {n} \} $
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runs through all regular sequences of intervals contracting towards the point $  x $.  
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Then the equation $  \overline{F}\; ( x) = \underline{F} ( x) $
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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 } \} $.  
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A sequence of intervals $  G _ {n} $
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is regular if there exist a number $  \alpha > 0 $
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and sequences of spheres $  S _ {n}  ^  \prime  $,  
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$  S _ {n}  ^ {\prime\prime} $
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such that for all $  n $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097090/w09709016.png" /></td> </tr></table>
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$$
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\mathop{\rm diam}  S _ {n}  ^  \prime  > \alpha  \mathop{\rm diam}  S _ {n}  ^ {\prime\prime}
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$$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097090/w09709017.png" /></td> </tr></table>
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$$
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S _ {n}  ^  \prime  \subset  G _ {n}  \subset  S _ {n}  ^ {\prime\prime} .
 +
$$
  
 
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 .
 
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 .
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====References====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Saks,  "Theory of the integral" , Hafner  (1952)  (Translated from French)</TD></TR></table>

Latest revision as of 08:28, 6 June 2020


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

Comments

References

[a1] S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)
How to Cite This Entry:
Ward theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ward_theorem&oldid=12355
This article was adapted from an original article by L.D. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article