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

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

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)