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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
 
An integral in which there is a successive integration with respect to different variables, i.e. an integral of the form
  
<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/r/r081/r081280/r0812801.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\int\limits _ {A _ {y} } \left [ \int\limits _ { A( } y) f( x, y) dx \right ]  dy.
 +
$$
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r0812802.png" /> is defined on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r0812803.png" /> lying in the direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r0812804.png" /> of spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r0812805.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r0812806.png" /> in which are given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r0812807.png" />-finite measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r0812808.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r0812809.png" /> and which have the completeness property; the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128010.png" /> (the  "section"  at  "level"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128011.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128012.png" />) is measurable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128013.png" />, while the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128014.png" /> (the projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128015.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128016.png" />) is measurable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128017.png" />. The integration over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128018.png" /> is performed with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128019.png" />, and that over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128020.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128021.png" />. The integral (1) is also denoted by
+
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
  
<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/r/r081/r081280/r08128022.png" /></td> </tr></table>
+
$$
 +
\int\limits _ {A _ {y} }  dy \int\limits _ { A( } y) f( x, y)  dx.
 +
$$
  
 
Multiple integrals (cf. [[Multiple integral|Multiple integral]]) can be reduced to repeated integrals.
 
Multiple integrals (cf. [[Multiple integral|Multiple integral]]) can be reduced to repeated integrals.
  
Let a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128023.png" />, integrable with respect to the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128024.png" /> on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128025.png" />, be extended by zero to a function on the entire space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128026.png" />. Then the 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
  
<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/r/r081/r081280/r08128027.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { Y }  dy \int\limits _ { X } f( x, y)  dx
 +
$$
  
 
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/r/r081/r081280/r08128028.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { X }  dx \int\limits _ { Y } f( x, y)  dy
 +
$$
  
 
exist and are equal to each other:
 
exist and are equal to each other:
  
<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/r/r081/r081280/r08128029.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\int\limits _ { Y }  dy \int\limits _ { X } f( x, y)  dx  = \
 +
\int\limits _ { X }  dx \int\limits _ { Y } f( x, y) dy
 +
$$
  
(see [[Fubini theorem|Fubini theorem]]). In the left-hand integral the outer integration is in fact performed over the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128030.png" />. In particular, for points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128031.png" /> the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128032.png" /> are measurable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128033.png" />. In general, one cannot take this integral over the entire set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128034.png" /> since, while the set A is measurable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128035.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128036.png" /> may be non-measurable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128037.png" />, and similarly, the individual sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128039.png" />, may be non-measurable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128040.png" />. On the other hand, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128041.png" /> is always measurable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128042.png" /> provided only that the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128043.png" /> is measurable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128044.png" />.
+
(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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128045.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128047.png" /> the repeated integrals are equal:
+
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:
  
<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/r/r081/r081280/r08128048.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { - } 1 ^ { + }  1  dx \int\limits _ { - } 1 ^ { + }  1 f( x, y)  dy  = \
 +
\int\limits _ { - } 1 ^ { + }  1  dy \int\limits _ { - } 1 ^ { + }  1 f( x, y)  dx  = 0,
 +
$$
  
 
while the multiple integral
 
while the multiple 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/r/r081/r081280/r08128049.png" /></td> </tr></table>
+
$$
 +
{\int\limits \int\limits } _ {| x | , | y | \leq  1 }
 +
f( x, y)  dx  dy
 +
$$
  
 
does not exist. However, if at least one of the integrals
 
does not exist. However, if at least one of the integrals
  
<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/r/r081/r081280/r08128050.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { Y }  dy \int\limits _ { X } | f( x, y) |  dx \  \textrm{ or } \ \
 +
\int\limits _ { X }  dx \int\limits _ { Y } | f( x, y) |  dy
 +
$$
  
is finite, then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128051.png" /> is integrable on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128052.png" /> and relation (2) holds.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128053.png" /> be summable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128054.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128055.png" /> for all values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128056.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128057.png" /> and let it be a function of bounded variation with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128058.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128059.png" /> for almost-all values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128060.png" />. Also, suppose that the total variation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128061.png" /> with respect to the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128062.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128063.png" /> for all given values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128064.png" /> does not exceed some non-negative summable function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128065.png" />. Then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128066.png" /> is a function of bounded variation with respect to the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128067.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128068.png" /> and for any continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128069.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081280/r08128070.png" /> one has the formula
+
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
  
<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/r/r081/r081280/r08128071.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { c } ^ { d }  dy \int\limits _ { a } ^ { b }  f( x)  d _ {x} g( x, y)  = \
 +
\int\limits _ { a } ^ { b }  f( x)  d _ {x} \left [ \int\limits _ { c } ^ { d }  g( x, y)  dy
 +
\right ] .
 +
$$
  
 
====References====
 
====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>
 
<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====
 
====Comments====

Revision as of 08:11, 6 June 2020


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) 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). 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 and the outer one is a 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

[1] V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian)
[2] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
[3] L.D. Kudryavtsev, "A course in mathematical analysis" , 2 , Moscow (1981) (In Russian)
[4] S.M. Nikol'skii, "A course of mathematical analysis" , 2 , MIR (1977) (Translated from Russian)
[5] V.I. Smirnov, "A course of higher mathematics" , 5 , Addison-Wesley (1964) (Translated from Russian)

Comments

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

References

[a1] E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965)
[a2] W. Rudin, "Real and complex analysis" , McGraw-Hill (1978) pp. 24
[a3] S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)
[a4] T.M. Apostol, "Mathematical analysis" , Addison-Wesley (1974)
[a5] P.R. Halmos, "Measure theory" , Springer (1974)
[a6] A.C. Zaanen, "Integration" , North-Holland (1974)
How to Cite This Entry:
Repeated integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Repeated_integral&oldid=19219
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article