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An integral in which there is a successive integration with respect to different variables, i.e. an integral of the form

$$ \tag{1 }
\int\limits _ {A _ {y} } \left [ \int\limits _ { A(  y)} f( x, y)  dx \right ]  dy.
$$

The function  $  f $
is defined on a set  $  A $
lying in the direct product  $  X \times Y $
of spaces  $  X $
and  $  Y $
in which are given  $  \sigma $-finite measures  $  \mu _ {x} $
and  $  \mu _ {y} $
and which have the completeness property; the set  $  A( y) = \{ {x } : {( x, y ) \in A } \} \subset  X $ (the  "section"  at  "level"   $  y \in Y $
of  $  A $) 
is measurable with respect to  $  \mu _ {x} $, 
while the set  $  A _ {y} $ (the projection of  $  A $
on  $  Y $) 
is measurable with respect to  $  \mu _ {y} $. 
The integration over  $  A ( y) $
is performed with respect to  $  \mu _ {x} $, 
and that over  $  A _ {y} $
with respect to  $  \mu _ {y} $. 
The integral (1) is also denoted by

$$ 
\int\limits _ {A _ {y} }  dy \int\limits _ { A(  y)} f( x, y)  dx.
$$

Multiple integrals (cf. [[Multiple integral|Multiple integral]]) can be reduced to repeated integrals.

Let a function  $  f $, 
integrable with respect to the measure  $  \mu = \mu _ {x} \times \mu _ {y} $
on the set  $  A \subset  X \times Y $, 
be extended by zero to a function on the entire space  $  X \times Y $. 
Then the repeated integrals

$$ 
\int\limits _ { Y }  dy \int\limits _ { X } f( x, y)  dx
$$

and

$$ 
\int\limits _ { X }  dx \int\limits _ { Y } f( x, y)  dy
$$

exist and are equal to each other:

$$ \tag{2 }
\int\limits _ { Y }  dy \int\limits _ { X } f( x, y)  dx  = \ 
\int\limits _ { X }  dx \int\limits _ { Y } f( x, y)  dy
$$

(see [[Fubini theorem|Fubini theorem]]). In the left-hand integral the outer integration is in fact performed over the set  $  A _ {y}  ^ {*} = \{ {y } : {y \in A _ {y} ,  \mu _ {x} A( y) > 0 } \} $. 
In particular, for points  $  y \in A _ {y}  ^ {*} $
the sets  $  A ( y) $
are measurable with respect to  $  \mu _ {x} $. 
In general, one cannot take this integral over the entire set  $  A _ {y} $
since, while the set A is measurable with respect to  $  \mu $, 
the set  $  A _ {y} $
may be non-measurable with respect to  $  \mu _ {y} $, 
and similarly, the individual sets  $  A ( y) $, 
$  y \in A _ {y} $, 
may be non-measurable with respect to  $  \mu _ {x} $. 
On the other hand, the set  $  A _ {y}  ^ {*} $
is always measurable with respect to  $  \mu _ {y} $
provided only that the set  $  A $
is measurable with respect to  $  \mu $.

The above conditions for changing the order of integration in a repeated integral are only sufficient but not necessary; sometimes it is permissible to change the order of integration in a repeated integral while the corresponding multiple integral does not exist. For example, for the function  $  f( x, y) = {xy / {( x  ^ {2} + y  ^ {2} )  ^ {2} } } $
for  $  x  ^ {2} + y  ^ {2} > 0 $
and  $  f( 0, 0) = 0 $
the repeated integrals are equal:

$$ 
\int\limits _ { -  1} ^ { +   1}  dx \int\limits _ { -  1} ^ { +   1} f( x, y)  dy  = \ 
\int\limits _ { -  1} ^ { +   1}  dy \int\limits _ { -  1} ^ { +   1} f( x, y)  dx  =  0,
$$

while the multiple integral

$$ 
{\int\limits \int\limits } _ {| x | , | y | \leq  1 }
f( x, y)  dx  dy
$$

does not exist. However, if at least one of the integrals

$$ 
\int\limits _ { Y }  dy \int\limits _ { X } | f( x, y) |  dx \  \textrm{ or } \ \ 
\int\limits _ { X }  dx \int\limits _ { Y } | f( x, y) |  dy
$$

is finite, then the function  $  f $
is integrable on the set  $  X \times Y $
and relation (2) holds.

In the case where the inner integral is a [[Stieltjes integral|Stieltjes integral]] and the outer one is a [[Lebesgue integral|Lebesgue integral]], the following theorem on changing the order of integration holds: Let a function  $  g( x, y) $
be summable with respect to  $  y $
in  $  [ c, d] $
for all values of  $  x $
in  $  [ a, b] $
and let it be a function of bounded variation with respect to  $  x $
in  $  [ a, b] $
for almost-all values  $  y \in [ c, d] $. 
Also, suppose that the total variation of  $  g $
with respect to the variable  $  x $
in  $  [ a, b] $
for all given values of  $  y $
does not exceed some non-negative summable function on  $  [ c, d] $. 
Then the function  $  \int _ {c}  ^ {d} g( x, y)  dy $
is a function of bounded variation with respect to the variable  $  x $
in  $  [ a, b] $
and for any continuous function  $  f $
on  $  [ a, b] $
one has the formula

$$ 
\int\limits _ { c } ^ { d }  dy \int\limits _ { a } ^ { b }  f( x)  d _ {x} g( x, y)  = \ 
\int\limits _ { a } ^ { b }  f( x)  d _ {x} \left [ \int\limits _ { c } ^ { d }  g( x, y)  dy
\right ] .
$$

====References====
<table><TR><TD valign="top">[1]</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">[2]</TD> <TD valign="top">  A.N. Kolmogorov,   S.V. Fomin,   "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.D. Kudryavtsev,   "A course in mathematical analysis" , '''2''' , Moscow  (1981)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.M. Nikol'skii,   "A course of mathematical analysis" , '''2''' , MIR  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.I. Smirnov,   "A course of higher mathematics" , '''5''' , Addison-Wesley  (1964)  (Translated from Russian)</TD></TR></table>

====Comments====
Instead of  "repeated integral"  one also uses iterated integral (cf., e.g., [[#References|[a1]]], [[#References|[a2]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Hewitt,   K.R. Stromberg,   "Real and abstract analysis" , Springer  (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Rudin,   "Real and complex analysis" , McGraw-Hill  (1978)  pp. 24</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Saks,   "Theory of the integral" , Hafner  (1952)  (Translated from French)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  T.M. Apostol,   "Mathematical analysis" , Addison-Wesley  (1974)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  P.R. Halmos,   "Measure theory" , Springer  (1974)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A.C. Zaanen,   "Integration" , North-Holland  (1974)</TD></TR></table>