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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 ] $:

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

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).

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d030160a.gif" />

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|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 [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Darboux,   ''Ann. Sci. Ecole Norm. Sup. Sér. 2'' , '''4'''  (1875)  pp. 57–112</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.A. Il'in,   E.G. Poznyak,   "Fundamentals of mathematical analysis" , '''1–2''' , MIR  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.D. Kudryavtsev,   "Mathematical analysis" , '''1–2''' , Moscow  (1973)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.M. Nikol'skii,   "A course of mathematical analysis" , '''1–2''' , MIR  (1977)  (Translated from Russian)</TD></TR></table>

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