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A generalization of the concept of the [[Riemann integral|Riemann integral]], realizing the notion of integrating a function  $  f $
with respect to another function  $  u $. 
Let two functions  $  f $
and  $  u $
be defined and bounded on  $  [ a, b] $
and let  $  a = x _ {0} < \dots < x _ {n} = b $. 
A sum of the form

$$ \tag{1 }
\sigma  =  f( \xi _ {1} ) [ u( x _ {1} ) - u( x _ {0} )] + \dots
+ f( \xi _ {n} )[ u( x _ {n} ) - u( x _ {n-} 1 )],
$$

where  $  x _ {i-} 1 \leq  \xi _ {i} \leq  x _ {i} $, 
$  i = 1 \dots n $, 
is called a Stieltjes integral sum. A number  $  I $
is called the limit of the integral sums (1) when  $  \max _ {i}  \Delta x _ {i} \rightarrow 0 $
if for each  $  \epsilon > 0 $
there is a  $  \delta > 0 $
such that if  $  \max  \Delta x _ {i} < \delta $, 
the inequality  $  | \sigma - I | < \epsilon $
holds. If the limit  $  I $
exists when  $  \max _ {i}  \Delta x _ {i} \rightarrow 0 $
and is finite, then the function  $  f $
is said to be integrable with respect to the function  $  u $
over  $  [ a, b] $, 
and the limit is called the Stieltjes integral (or the Riemann–Stieltjes integral) of  $  f $
with respect to  $  u $, 
and is denoted by

$$ \tag{2 }
I  =  \int\limits _ { a } ^ { b }  f( x)  du( x);
$$

the function  $  u $
is said to be the integrating function. Th.J. Stieltjes [[#References|[1]]] hit upon the idea of such an integral when studying the positive  "distribution of masses"  on a straight line defined by an increasing function  $  u $, 
the points of discontinuity of which correspond to masses that are  "concentrated at one point" .

The Riemann integral is a particular case of the Stieltjes integral, when a function  $  x+ C $, 
where  $  C = \textrm{ const } $, 
is taken as the integrating function  $  u $.

When the integrating function  $  u $
increases monotonically, the upper and lower Darboux–Stieltjes sums are studied:

$$ \tag{3 }
S  =  \sum _ { i= } 1 ^ { n }  M _ {i} [ u( x _ {i} ) - u( x _ {i-} 1 )],
$$

$$ 
s  =  \sum _ { i= } 1 ^ { n }  m _ {i} [ u( x _ {i} ) - u( x _ {i-} 1 )],
$$

where  $  m _ {i} $
and  $  M _ {i} $
are the greatest lower and least upper bounds of  $  f $
on  $  [ x _ {i-} 1 , x _ {i} ] $.

For a Stieltjes integral to exists, it is sufficient for one of the following conditions to be fulfilled:

1) the function  $  f $
is continuous on  $  [ a, b] $, 
while the function  $  u $
is of bounded variation on  $  [ a, b] $;

2) the function  $  f $
is Riemann integrable on  $  [ a, b] $, 
while the function  $  u $
satisfies a Lipschitz condition on  $  [ a, b] $, 
i.e.  $  | u( x _ {1} ) - u( x _ {2} ) | \leq  C  | x _ {1} - x _ {2} | $, 
where  $  C = \textrm{ const } $, 
for any  $  x _ {1} $
and  $  x _ {2} $
from  $  [ a, b] $;

3) the function  $  f $
is Riemann integrable on  $  [ a, b] $, 
while the function  $  u $
can be represented on  $  [ a, b] $
as an integral with a variable upper bound,

$$ 
u( x)  =  C + \int\limits _ { a } ^ { x }  g( t)  dt,
$$

where  $  g $
is absolutely integrable over  $  a \leq  t \leq  b $.

When condition 3) is fulfilled, the integral (2) reduces to a [[Lebesgue integral|Lebesgue integral]] by the formula

$$ \tag{4 }
\int\limits _ { a } ^ { b }  f( x)  du( x)  = \ 
\int\limits _ { a } ^ { b }  f( x) g( x)  dx.
$$

(The right-hand side is a Riemann integral if  $  g $
is Riemann integrable.) In particular, (4) holds if  $  u $
has a bounded and Riemann-integrable derivative  $  u  ^  \prime  $
on  $  [ a, b] $; 
in this case  $  g = u  ^  \prime  $.

If  $  u $
is integrable with respect to  $  f $
over  $  [ a, b] $, 
then  $  f $
is also integrable with respect to  $  u $
over  $  [ a, b] $. 
This statement leads to a number of further conditions on the existence of Stieltjes integrals.

The Stieltjes integral has the property of linearity relative to both the integrand and the integrating function (given the condition that every one of the Stieltjes integrals on the right-hand side exists):

$$ 
\int\limits _ { a } ^ { b }  [ \alpha f _ {1} ( x) + \beta f _ {2} ( x)]  du( x) =
$$

$$ 
= \ 
\alpha \int\limits _ { a } ^ { b }  f _ {1} ( x)  du( x) + \beta \int\limits _ { a } ^ { b }  f _ {2} ( x)  du( x),
$$

$$ 
\int\limits _ { a } ^ { b }  f( x)  d[ \alpha u _ {1} ( x) + \beta u _ {2} ( x)] =
$$

$$ 
= \ 
\alpha \int\limits _ { a } ^ { b }  f( x)  du _ {1} ( x) + \beta \int\limits _ { a } ^ { b }  f( x)  du _ {2} ( x).
$$

Generally speaking, Stieltjes integrals do not possess the property of additivity: The existence of  $  \int _ {a}  ^ {b} f( x)  du( x) $
does not follow from the existence of both the integrals  $  \int _ {a}  ^ {c} f( x)  du( x) $
and  $  \int _ {c}  ^ {b} f( x)  du( x) $(
the converse is, instead, true if  $  a < c < b $).

If  $  f $
is bounded on  $  [ a, b] $, 
$  m \leq  f( x) \leq  M $, 
and  $  u $
increases monotonically on  $  [ a, b] $, 
then there exists a  $  \mu $
satisfying the inequality  $  m \leq  \mu \leq  M $, 
such that the mean-value formula

$$ \tag{5 }
\int\limits _ { a } ^ { b }  f( x)  du( x)  = \ 
\mu [ u( b) - u( a)]
$$

holds for a Stieltjes integral. In particular, if  $  f $
is continuous on  $  [ a, b] $, 
then there exists a point  $  \xi \in [ a, b] $
such that  $  \mu = f( \xi ) $.

A Stieltjes integral  $  \int _ {a}  ^ {b} f( x)  du( x) $, 
where  $  u $
is of bounded variation, provides the general form of a continuous linear functional  $  F( f  ) $
on the space of continuous functions on  $  [ a, b] $(
Riesz' theorem).

When the function  $  u $
is of bounded variation, the value of the Stieltjes integral coincides with the value of the corresponding [[Lebesgue–Stieltjes integral|Lebesgue–Stieltjes integral]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Th.J. Stieltjes,   "Recherches sur les fractions continues"  ''C.R. Acad. Sci. Paris'' , '''118'''  (1894)  pp. 1401–1403</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.I. Smirnov,   "A course of higher mathematics" , '''5''' , Addison-Wesley  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.I. Glivenko,   "The Stieltjes integral" , Moscow-Leningrad  (1936)  (In Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K.A. Ross,   "Elementary analysis: The theory of calculus" , Springer  (1980)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Rudin,   "Principles of mathematical analysis" , McGraw-Hill  (1976)  pp. 75–78</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T.M. Apostol,   "Mathematical analysis" , Addison-Wesley  (1974)</TD></TR></table>