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A sum of special type. Let a real function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d0301601.png" /> be defined and bounded on a segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d0301602.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d0301603.png" /> be a decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d0301604.png" />:
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<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/d/d030/d030160/d0301605.png" /></td> </tr></table>
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 +
A sum of special type. Let a real function  $  f $
 +
be defined and bounded on a segment  $  [ a , b ] $,
 +
let  $  \tau = {\{ x _ {i} \} } _ {i=} 0  ^ {k} $
 +
be a decomposition of  $  [ a , b ] $:
 +
 
 +
$$
 +
= x _ {0< x _ {1}  < \dots < x _ {k}  =  b ,
 +
$$
  
 
and set
 
and set
  
<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/d/d030/d030160/d0301606.png" /></td> </tr></table>
+
$$
 +
m _ {i}  = \
 +
\inf _ {x _ {i-} 1 \leq  x \leq  x _ {i} }  f( x),\ \
 +
M _ {i}  =   \sup  _ {x _ {i-} 1 \leq  x \leq  x _ {i} }  f ( x) ,
 +
$$
  
<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/d/d030/d030160/d0301607.png" /></td> </tr></table>
+
$$
 +
\Delta x _ {i}  = x _ {i} - x _ {i-} 1 ,\  i = 1 \dots k .
 +
$$
  
 
The sums
 
The sums
  
<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/d/d030/d030160/d0301608.png" /></td> </tr></table>
+
$$
 +
s _  \tau  = \sum _ {i = 1 } ^ { k }  m _ {i} \Delta x _ {i} \ 
 +
\textrm{ and } \ \
 +
S _  \tau  = \sum _ {i = 1 } ^ { k }  M _ {i} \Delta x _ {i}  $$
  
are known, respectively, as the lower and upper Darboux sums. For any two decompositions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d0301609.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016011.png" /> the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016012.png" /> is valid, i.e. any lower Darboux sum is no larger than an upper. If
+
are known, respectively, as the lower and upper Darboux sums. For any two decompositions $  \tau $
 +
and $  \tau  ^  \prime  $
 +
of $  [ a , b ] $
 +
the inequality $  s _  \tau  \leq  S _ {\tau  ^  \prime  } $
 +
is valid, i.e. any lower Darboux sum is no larger than an upper. If
  
<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/d/d030/d030160/d03016013.png" /></td> </tr></table>
+
$$
 +
\sigma _  \tau  = \sum _ {i = 1 } ^ { k }  f ( \xi _ {i} ) \Delta x _ {i} ,
 +
\  \xi _ {i} \in [ x _ {i-} 1 , x _ {i} ] ,
 +
$$
  
 
is a Riemann sum, then
 
is a Riemann sum, then
  
<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/d/d030/d030160/d03016014.png" /></td> </tr></table>
+
$$
 +
s _  \tau  = \inf _ {\xi _ {1} \dots \xi _ {k} }  \sigma _  \tau  ,\  S _  \tau  =   \sup  _ {\xi _ {1} \dots \xi _ {k} }  \sigma _  \tau  .
 +
$$
  
The geometric meaning of the lower and upper Darboux sums is that they are equal to the planar areas of stepped figures consisting of rectangles whose base widths are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016015.png" /> and with respective heights <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016017.png" /> (see Fig.) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016018.png" />. These figures approximate, from the inside and outside, the curvilinear trapezium formed by the graph of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016019.png" />, the abscissa axis and the rectilinear segments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016021.png" /> (which may degenerate into points).
+
The geometric meaning of the lower and upper Darboux sums is that they are equal to the planar areas of stepped figures consisting of rectangles whose base widths are $  \Delta x _ {i} $
 +
and with respective heights $  m _ {i} $
 +
and $  M _ {i} $(
 +
see Fig.) if $  f \geq  0 $.  
 +
These figures approximate, from the inside and outside, the curvilinear trapezium formed by the graph of $  f $,  
 +
the abscissa axis and the rectilinear segments $  x = a $
 +
and $  x= b $(
 +
which may degenerate into points).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d030160a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d030160a.gif" />
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The numbers
 
The numbers
  
<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/d/d030/d030160/d03016022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
I _ {*}  = \sup _  \tau  s _  \tau  ,\ \
 +
I  ^ {*}  = \inf _  \tau  S _  \tau  $$
  
are called, respectively, the lower and the upper Darboux integrals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016023.png" />. They are the limits of the lower and the upper Darboux sums:
+
are called, respectively, the lower and the upper Darboux integrals of $  f $.  
 +
They are the limits of the lower and the upper Darboux sums:
  
<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/d/d030/d030160/d03016024.png" /></td> </tr></table>
+
$$
 +
I _ {*}  = \lim\limits _ {\delta _  \tau  \rightarrow 0 }  s _  \tau  ,\ \
 +
I  ^ {*}  = \lim\limits _ {\delta _  \tau  \rightarrow 0 }  S _  \tau  ,
 +
$$
  
 
where
 
where
  
<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/d/d030/d030160/d03016025.png" /></td> </tr></table>
+
$$
 +
\delta _  \tau  = \max _ {i = 1 \dots k }  \Delta x _ {i}  $$
  
is the fineness (mesh) of the decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016026.png" />. The condition
+
is the fineness (mesh) of the decomposition $  \tau $.  
 +
The condition
  
<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/d/d030/d030160/d03016027.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
I _ {*}  = I  ^ {*}
 +
$$
  
is necessary and sufficient for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016028.png" /> to be Riemann integrable on the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016029.png" />. Here, if condition (2) is met, the value of the lower and the upper Darboux integrals becomes identical with the Riemann integral
+
is necessary and sufficient for a function $  f $
 +
to be Riemann integrable on the segment $  [ a , b ] $.  
 +
Here, if condition (2) is met, the value of the lower and the upper Darboux integrals becomes identical with the Riemann integral
  
<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/d/d030/d030160/d03016030.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { a } ^ { b }  f ( x)  dx .
 +
$$
  
With the aid of Darboux sums, condition (2) may be formulated in the following equivalent form: For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016031.png" /> there exists a decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016032.png" /> such that
+
With the aid of Darboux sums, condition (2) may be formulated in the following equivalent form: For each $  \epsilon > 0 $
 +
there exists a decomposition $  \tau $
 +
such that
  
<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/d/d030/d030160/d03016033.png" /></td> </tr></table>
+
$$
 +
S _  \tau  - s _  \tau  < \epsilon .
 +
$$
  
 
The condition
 
The condition
  
<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/d/d030/d030160/d03016034.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {\delta _  \tau  \rightarrow 0 } ( S _  \tau  - s _  \tau  )  = 0
 +
$$
  
is also necessary and sufficient for the Riemann integrability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016035.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016036.png" />. Here
+
is also necessary and sufficient for the Riemann integrability of $  f $
 +
on $  [ a , b ] $.  
 +
Here
  
<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/d/d030/d030160/d03016037.png" /></td> </tr></table>
+
$$
 +
S _  \tau  - s _  \tau  = \sum _ {i = 1 } ^ { k }
 +
\omega _ {i} ( f  ) \Delta x _ {i} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016038.png" /> is the oscillation (cf. [[Oscillation of a function|Oscillation of a function]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016039.png" /> on
+
where $  \omega _ {i} ( f  ) $
 +
is the oscillation (cf. [[Oscillation of a function|Oscillation of a function]]) of $  f $
 +
on
  
<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/d/d030/d030160/d03016040.png" /></td> </tr></table>
+
$$
 +
[ x _ {i-} 1 , x _ {i} ] ,\  i = 1 \dots k .
 +
$$
  
The concept of lower and upper Darboux sums may be generalized to the case of functions of several variables which are measurable in the sense of some positive measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016041.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016042.png" /> be a measurable (for example, Jordan or Lebesgue) subset of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016043.png" />-dimensional space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016044.png" /> and suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016045.png" /> is finite. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016046.png" /> be a decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016047.png" />, i.e. a system of measurable subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016048.png" /> such that
+
The concept of lower and upper Darboux sums may be generalized to the case of functions of several variables which are measurable in the sense of some positive measure $  \mu $.  
 +
Let $  E $
 +
be a measurable (for example, Jordan or Lebesgue) subset of the $  n $-
 +
dimensional space, $  n = 1 , 2 \dots $
 +
and suppose $  \mu ( E) $
 +
is finite. Let $  \tau = \{ E _ {i} \} _ {i=} 1  ^ {k} $
 +
be a decomposition of $  E $,  
 +
i.e. a system of measurable subsets of $  E $
 +
such that
  
<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/d/d030/d030160/d03016049.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
\cup _ {i= 1 } ^ { k }  E _ {i}  = E ,
 +
$$
  
<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/d/d030/d030160/d03016050.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
\mu ( E _ {i} \cap E _ {j }  )  = 0 \  \textrm{ if }  i \neq j .
 +
$$
  
Let a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016051.png" /> be bounded on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016052.png" /> and let
+
Let a function $  f $
 +
be bounded on $  E $
 +
and let
  
<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/d/d030/d030160/d03016053.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
$$ \tag{5 }
 +
m _ {i}  = \inf _ {x \in E _ {i} }  f ( x),
 +
\  M _ {i}  = \sup
 +
_ {x \in E _ {i} }  f ( x) ,\  i = 1 \dots k .
 +
$$
  
 
The sums
 
The sums
  
<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/d/d030/d030160/d03016054.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>
+
$$ \tag{6 }
 +
s _  \tau  = \sum _ {i = 1 } ^ { k }  m _ {i} \mu ( E _ {i} ) ,\ \
 +
S _  \tau  = \sum _ {i = 1 } ^ { k }  M _ {i} \mu ( E _ {i} )
 +
$$
  
are also said to be, respectively, lower and upper Darboux sums. The lower <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016055.png" /> and the upper <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016056.png" /> integrals are defined by formulas (1). For Jordan measure, their equality is a sufficient and necessary condition for the function to be Riemann integrable, and their common value coincides with the Riemann integral. For Lebesgue measure, on the other hand, the equality
+
are also said to be, respectively, lower and upper Darboux sums. The lower $  I _ {*} $
 +
and the upper $  I  ^ {*} $
 +
integrals are defined by formulas (1). For Jordan measure, their equality is a sufficient and necessary condition for the function to be Riemann integrable, and their common value coincides with the Riemann integral. For Lebesgue measure, on the other hand, the equality
  
<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/d/d030/d030160/d03016057.png" /></td> </tr></table>
+
$$
 +
I _ {*}  = I  ^ {*}  = \int\limits _ { E } f( x)  dx.
 +
$$
  
 
is always valid for bounded Lebesgue-measurable functions.
 
is always valid for bounded Lebesgue-measurable functions.
  
In general, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016058.png" /> is a complete <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016059.png" />-additive bounded measure, defined on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016060.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016061.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016062.png" /> is a bounded measurable real-valued function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016063.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016064.png" /> is a decomposition of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016065.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016066.png" />-measurable sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016067.png" /> which satisfy the conditions (3) and (4), and if the Darboux sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016069.png" /> are defined by formulas (5) and (6), while the integrals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016070.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016071.png" /> are defined by the formulas (1), in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016072.png" /> is always understood to mean the measure under consideration, then
+
In general, if $  \mu $
 +
is a complete $  \sigma $-
 +
additive bounded measure, defined on a $  \sigma $-
 +
algebra $  \mathfrak S _  \mu  $,  
 +
if $  f $
 +
is a bounded measurable real-valued function on $  E $,  
 +
if $  \tau = \{ E _ {i} \} _ {i= 1 }  ^ {k} $
 +
is a decomposition of a set $  E \in \mathfrak S _  \mu  $
 +
into $  \mu $-
 +
measurable sets $  E _ {i} $
 +
which satisfy the conditions (3) and (4), and if the Darboux sums $  s _  \tau  $
 +
and $  S _  \tau  $
 +
are defined by formulas (5) and (6), while the integrals $  I _ {*} $
 +
and $  I  ^ {*} $
 +
are defined by the formulas (1), in which $  \mu $
 +
is always understood to mean the measure under consideration, then
  
<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/d/d030/d030160/d03016073.png" /></td> </tr></table>
+
$$
 +
I _ {*}  = I  ^ {*}  = \int\limits _ { E } f( x)  d \mu .
 +
$$
  
A generalization of the Darboux sums to unbounded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016074.png" />-measurable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016075.png" /> defined on sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016076.png" /> are the series (if they are absolutely convergent)
+
A generalization of the Darboux sums to unbounded $  \mu $-
 +
measurable functions $  f $
 +
defined on sets $  E \in \mathfrak S _  \mu  $
 +
are the series (if they are absolutely convergent)
  
<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/d/d030/d030160/d03016077.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>
+
$$ \tag{7 }
 +
s _  \tau  = \sum _ { i= } 1 ^  \infty  m _ {i} \mu ( E _ {i} ) ,\ \
 +
S _  \tau  = \sum _ { i= } 1 ^  \infty  M _ {i} \mu ( E _ {i} )
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016078.png" /> is a decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016079.png" /> (this decomposition consists, generally speaking, of an infinite number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016080.png" />-measurable sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016081.png" /> which satisfy condition (4) and are, of course, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016082.png" />), while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016083.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016084.png" /> are defined by (5). In (7) (as in (6) above) it is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016085.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016086.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016087.png" /> are again defined according to (1) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016088.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016089.png" /> are now defined in the sense of (7) and exist for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016090.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016091.png" />. If the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016092.png" /> is finite, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016093.png" /> is integrable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016094.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030160/d03016095.png" />.
+
where $  \tau = \{ E _ {i} \} _ {i=} 1  ^  \infty  $
 +
is a decomposition of $  E \in \mathfrak S _  \mu  $(
 +
this decomposition consists, generally speaking, of an infinite number of $  \mu $-
 +
measurable sets $  E _ {i} $
 +
which satisfy condition (4) and are, of course, such that $  \cup _ {i=} 1  ^  \infty  E _ {i} = E $),  
 +
while $  m _ {i} $
 +
and $  M _ {i} $
 +
are defined by (5). In (7) (as in (6) above) it is assumed that $  \infty \cdot 0 = 0 \cdot \infty = 0 $.  
 +
If $  I _ {*} $
 +
and $  I  ^ {*} $
 +
are again defined according to (1) and $  s _  \tau  $
 +
and $  S _  \tau  $
 +
are now defined in the sense of (7) and exist for each $  \tau $,  
 +
then $  I _ {*} = I  ^ {*} $.  
 +
If the value $  I = I _ {*} = I  ^ {*} $
 +
is finite, then $  f $
 +
is integrable with respect to $  \mu $
 +
and $  I = \int _ {E} f ( x)  d \mu $.
  
 
Named after G. Darboux [[#References|[1]]].
 
Named after G. Darboux [[#References|[1]]].

Latest revision as of 17:32, 5 June 2020


A sum of special type. Let a real function $ f $ be defined and bounded on a segment $ [ a , b ] $, let $ \tau = {\{ x _ {i} \} } _ {i=} 0 ^ {k} $ be a decomposition of $ [ a , b ] $:

$$ a = x _ {0} < x _ {1} < \dots < x _ {k} = b , $$

and set

$$ m _ {i} = \ \inf _ {x _ {i-} 1 \leq x \leq x _ {i} } f( x),\ \ M _ {i} = \sup _ {x _ {i-} 1 \leq x \leq x _ {i} } f ( x) , $$

$$ \Delta x _ {i} = x _ {i} - x _ {i-} 1 ,\ i = 1 \dots k . $$

The sums

$$ s _ \tau = \sum _ {i = 1 } ^ { k } m _ {i} \Delta x _ {i} \ \textrm{ and } \ \ S _ \tau = \sum _ {i = 1 } ^ { k } M _ {i} \Delta x _ {i} $$

are known, respectively, as the lower and upper Darboux sums. For any two decompositions $ \tau $ and $ \tau ^ \prime $ of $ [ a , b ] $ the inequality $ s _ \tau \leq S _ {\tau ^ \prime } $ is valid, i.e. any lower Darboux sum is no larger than an upper. If

$$ \sigma _ \tau = \sum _ {i = 1 } ^ { k } f ( \xi _ {i} ) \Delta x _ {i} , \ \xi _ {i} \in [ x _ {i-} 1 , x _ {i} ] , $$

is a Riemann sum, then

$$ s _ \tau = \inf _ {\xi _ {1} \dots \xi _ {k} } \sigma _ \tau ,\ S _ \tau = \sup _ {\xi _ {1} \dots \xi _ {k} } \sigma _ \tau . $$

The geometric meaning of the lower and upper Darboux sums is that they are equal to the planar areas of stepped figures consisting of rectangles whose base widths are $ \Delta x _ {i} $ and with respective heights $ m _ {i} $ and $ M _ {i} $( see Fig.) if $ f \geq 0 $. These figures approximate, from the inside and outside, the curvilinear trapezium formed by the graph of $ f $, the abscissa axis and the rectilinear segments $ x = a $ and $ x= b $( which may degenerate into points).

Figure: d030160a

The numbers

$$ \tag{1 } I _ {*} = \sup _ \tau s _ \tau ,\ \ I ^ {*} = \inf _ \tau S _ \tau $$

are called, respectively, the lower and the upper Darboux integrals of $ f $. They are the limits of the lower and the upper Darboux sums:

$$ I _ {*} = \lim\limits _ {\delta _ \tau \rightarrow 0 } s _ \tau ,\ \ I ^ {*} = \lim\limits _ {\delta _ \tau \rightarrow 0 } S _ \tau , $$

where

$$ \delta _ \tau = \max _ {i = 1 \dots k } \Delta x _ {i} $$

is the fineness (mesh) of the decomposition $ \tau $. The condition

$$ \tag{2 } I _ {*} = I ^ {*} $$

is necessary and sufficient for a function $ f $ to be Riemann integrable on the segment $ [ a , b ] $. Here, if condition (2) is met, the value of the lower and the upper Darboux integrals becomes identical with the Riemann integral

$$ \int\limits _ { a } ^ { b } f ( x) dx . $$

With the aid of Darboux sums, condition (2) may be formulated in the following equivalent form: For each $ \epsilon > 0 $ there exists a decomposition $ \tau $ such that

$$ S _ \tau - s _ \tau < \epsilon . $$

The condition

$$ \lim\limits _ {\delta _ \tau \rightarrow 0 } ( S _ \tau - s _ \tau ) = 0 $$

is also necessary and sufficient for the Riemann integrability of $ f $ on $ [ a , b ] $. Here

$$ S _ \tau - s _ \tau = \sum _ {i = 1 } ^ { k } \omega _ {i} ( f ) \Delta x _ {i} , $$

where $ \omega _ {i} ( f ) $ is the oscillation (cf. Oscillation of a function) of $ f $ on

$$ [ x _ {i-} 1 , x _ {i} ] ,\ i = 1 \dots k . $$

The concept of lower and upper Darboux sums may be generalized to the case of functions of several variables which are measurable in the sense of some positive measure $ \mu $. Let $ E $ be a measurable (for example, Jordan or Lebesgue) subset of the $ n $- dimensional space, $ n = 1 , 2 \dots $ and suppose $ \mu ( E) $ is finite. Let $ \tau = \{ E _ {i} \} _ {i=} 1 ^ {k} $ be a decomposition of $ E $, i.e. a system of measurable subsets of $ E $ such that

$$ \tag{3 } \cup _ {i= 1 } ^ { k } E _ {i} = E , $$

$$ \tag{4 } \mu ( E _ {i} \cap E _ {j } ) = 0 \ \textrm{ if } i \neq j . $$

Let a function $ f $ be bounded on $ E $ and let

$$ \tag{5 } m _ {i} = \inf _ {x \in E _ {i} } f ( x), \ M _ {i} = \sup _ {x \in E _ {i} } f ( x) ,\ i = 1 \dots k . $$

The sums

$$ \tag{6 } s _ \tau = \sum _ {i = 1 } ^ { k } m _ {i} \mu ( E _ {i} ) ,\ \ S _ \tau = \sum _ {i = 1 } ^ { k } M _ {i} \mu ( E _ {i} ) $$

are also said to be, respectively, lower and upper Darboux sums. The lower $ I _ {*} $ and the upper $ I ^ {*} $ integrals are defined by formulas (1). For Jordan measure, their equality is a sufficient and necessary condition for the function to be Riemann integrable, and their common value coincides with the Riemann integral. For Lebesgue measure, on the other hand, the equality

$$ I _ {*} = I ^ {*} = \int\limits _ { E } f( x) dx. $$

is always valid for bounded Lebesgue-measurable functions.

In general, if $ \mu $ is a complete $ \sigma $- additive bounded measure, defined on a $ \sigma $- algebra $ \mathfrak S _ \mu $, if $ f $ is a bounded measurable real-valued function on $ E $, if $ \tau = \{ E _ {i} \} _ {i= 1 } ^ {k} $ is a decomposition of a set $ E \in \mathfrak S _ \mu $ into $ \mu $- measurable sets $ E _ {i} $ which satisfy the conditions (3) and (4), and if the Darboux sums $ s _ \tau $ and $ S _ \tau $ are defined by formulas (5) and (6), while the integrals $ I _ {*} $ and $ I ^ {*} $ are defined by the formulas (1), in which $ \mu $ is always understood to mean the measure under consideration, then

$$ I _ {*} = I ^ {*} = \int\limits _ { E } f( x) d \mu . $$

A generalization of the Darboux sums to unbounded $ \mu $- measurable functions $ f $ defined on sets $ E \in \mathfrak S _ \mu $ are the series (if they are absolutely convergent)

$$ \tag{7 } s _ \tau = \sum _ { i= } 1 ^ \infty m _ {i} \mu ( E _ {i} ) ,\ \ S _ \tau = \sum _ { i= } 1 ^ \infty M _ {i} \mu ( E _ {i} ) $$

where $ \tau = \{ E _ {i} \} _ {i=} 1 ^ \infty $ is a decomposition of $ E \in \mathfrak S _ \mu $( this decomposition consists, generally speaking, of an infinite number of $ \mu $- measurable sets $ E _ {i} $ which satisfy condition (4) and are, of course, such that $ \cup _ {i=} 1 ^ \infty E _ {i} = E $), while $ m _ {i} $ and $ M _ {i} $ are defined by (5). In (7) (as in (6) above) it is assumed that $ \infty \cdot 0 = 0 \cdot \infty = 0 $. If $ I _ {*} $ and $ I ^ {*} $ are again defined according to (1) and $ s _ \tau $ and $ S _ \tau $ are now defined in the sense of (7) and exist for each $ \tau $, then $ I _ {*} = I ^ {*} $. If the value $ I = I _ {*} = I ^ {*} $ is finite, then $ f $ is integrable with respect to $ \mu $ and $ I = \int _ {E} f ( x) d \mu $.

Named after G. Darboux [1].

References

[1] G. Darboux, Ann. Sci. Ecole Norm. Sup. Ser. 2 , 4 (1875) pp. 57–112
[2] V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian)
[3] L.D. Kudryavtsev, "Mathematical analysis" , 1–2 , Moscow (1973) (In Russian)
[4] S.M. Nikol'skii, "A course of mathematical analysis" , 1–2 , MIR (1977) (Translated from Russian)
How to Cite This Entry:
Darboux sum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Darboux_sum&oldid=18687
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article