# Repeated integral

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

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

How to Cite This Entry:
Repeated integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Repeated_integral&oldid=52306
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article