Multiple integral

A definite integral of a function of several variables. There are several different concepts of a multiple integral (Riemann integral, Lebesgue integral, Lebesgue–Stieltjes integral, etc.).

The multiple Riemann integral is based on the concept of a Jordan measure $ \mu $. Let $ E $ be a Jordan-measurable set in the $ n $- dimensional Euclidean space $ \mathbf R ^ {n} $, let $ \mu _ {n} $ be the $ n $- dimensional Jordan measure and let $ \tau = \{ E _ {i} \} _ {i = 1 } ^ {k} $ be a partition of $ E $, i.e. a system of Jordan-measurable sets $ E _ {i} $ such that $ \cup _ {i = 1 } ^ {k} E _ {i} = E $ and $ \mu _ {n} ( E _ {i} \cap E _ {j} ) = 0 $, $ i \neq j $, $ i, j = 1 \dots n $. The quantity

$$ \delta _ \tau = \max _ {i = 1 \dots k } d ( E _ {i} ), $$

where $ d ( E _ {i} ) $ is the diameter of $ E _ {i} $, is called the mesh of the partition $ \tau $. If $ f ( x) $, $ x = ( x _ {1} \dots x _ {n} ) $, is a function defined on $ E $, then any sum of the type

$$ \sigma _ \tau = \ \sigma _ \tau ( f; \xi ^ {(1)} \dots \xi ^ {(k)} ) = \ \sum _ {i = 1 } ^ { k } f ( \xi ^ {(i)} ) \mu _ {n} ( E _ {i} ), $$

$$ \xi ^ {(i)} \in E _ {i} \in \tau , $$

is called a Riemann integral sum of the function $ f $. If $ f $ has the property that $ \lim\limits _ {\delta _ \tau \rightarrow 0 } \sigma _ \tau $ exists, independently of the specific sequence of partitions, then this limit is called the $ n $- tuple Riemann integral of $ f $ over $ E $, and is denoted by

$$ \int\limits _ { E } f ( x) dx $$

or

$$ {\int\limits \dots \int\limits } _ { E } f ( x _ {1} \dots x _ {n} ) dx _ {1} \dots dx _ {n} . $$

The function $ f $ itself is then said to be Riemann integrable or, more briefly, R-integrable.

When $ n = 1 $, the set $ E $ over which the integration takes place is usually an interval and $ \tau $ is a partition consisting exclusively of intervals (see Riemann integral). Hence both the set over which the integration is performed and the elements of the partition are Jordan-measurable sets of a very special form — intervals. That is why not all the properties of functions which are R-integrable on an interval are valid for functions which are R-integrable on arbitrary Jordan-measurable sets. For example, since any function defined on a set of Jordan measure zero is R-integrable on that set, it follows that R-integrable functions need not be bounded. This is impossible for R-integrable functions on intervals. If one wishes R-integrability of a function on some set to imply that the function is bounded, certain additional conditions must be imposed on the set; for example, one might require that the set have arbitrarily fine partitions all elements of which have positive Jordan measure. The class defined by this condition includes all Jordan-measurable open sets and their closures, in particular all Jordan-measurable open domains and their closures. These are precisely the sets for which multiple Riemann integrals are most often used. When $ n = 2 $( $ n = 3 $), a multiple integral is called a double (triple) integral (cf. also Double integral).

Since a multiple Riemann integral can be evaluated only over Jordan-measurable sets (if $ n = 2 $ such a set is also called squarable; if $ n = 3 $ it is also called cubable), double (triple) Riemann integrals are considered only on sets (usually domains or closures of domains) with boundaries of Jordan area (volume) zero.

The Riemann integral of a bounded function of $ n $ variables ( $ n \geq 1 $) possesses the usual properties of an integral (linearity, additivity with respect to the set of integration, preservation of non-strict inequalities under integration, integrability of the product of integrable functions, etc.).

A multiple Riemann integral can be reduced to a repeated integral. Let $ x = ( x ^ \prime , x ^ {\prime\prime} ) \in \mathbf R ^ {n} $,

$$ x ^ \prime = ( x _ {1} \dots x _ {m} ) \in \mathbf R ^ {m} , $$

$$ x ^ {\prime\prime} = ( x _ {m + 1 } \dots x _ {n} ) \in \mathbf R ^ {n - m } ,\ E \subset \mathbf R ^ {n} , $$

where $ E $ is a Jordan-measurable set in $ \mathbf R ^ {n} $, $ E ( x _ {0} ^ \prime ) = E \cap \{ x ^ \prime = x _ {0} ^ \prime \} $ is the intersection of $ E $ with the $ ( n - m) $- dimensional hyperplane $ x ^ \prime = x _ {0} ^ \prime $, $ E _ {x ^ {\prime\prime} } $ is the projection of $ E $ on the hyperplane $ \mathbf R ^ {m} = \{ {x } : {x ^ {\prime\prime} = 0 } \} $, with $ E ( x ^ \prime ) $ and $ E _ {x ^ {\prime\prime} } $ measurable in the sense of the $ ( n - m) $- dimensional and $ m $- dimensional Jordan measure, respectively. If $ f $ is an R-integrable function on $ E $ and if for all $ x ^ \prime \in E _ {x ^ {\prime\prime} } $ the $ ( n - m) $- multiple integrals of the restrictions of $ f $ to the set $ E ( x ^ \prime ) $ exist, then the repeated integral

$$ \int\limits _ {E _ {x ^ {\prime\prime} } } dx ^ \prime \int\limits _ {E ( x ^ \prime ) } f ( x ^ \prime , x ^ {\prime\prime} ) dx ^ {\prime\prime} , $$

where the outer integral is an $ m $- tuple Riemann integral, exists, and

$$ \int\limits _ { E } f ( x) dx = \ {\int\limits \int\limits } _ { E } f ( x ^ \prime , x ^ {\prime\prime} ) dx ^ \prime dx ^ {\prime\prime\ } = $$

$$ = \ \int\limits _ {E _ {x ^ {\prime\prime} } } dx ^ \prime \int\limits _ {E ( x ^ \prime ) } f ( x ^ \prime , x ^ {\prime\prime} ) dx ^ {\prime\prime} . $$

For $ n = 3 $ this implies the following formulas:

1) If $ E \subset \mathbf R _ {xyz} ^ {3} $, if $ E _ {xy} $ is the projection of $ E $ on the $ xy $- plane, and if $ \phi ( x, y) $ and $ \psi ( x, y) $, $ x, y \in E _ {xy} $, are functions with graphs bounded by the set $ E $ in the $ z $- direction, i.e.

$$ E = \{ {( x, y, z) } : {( x, y) \in E _ {xy} ,\ \phi ( x, y) \leq z \leq \psi ( x, y) } \} , $$

then

$$ {\int\limits \int\limits \int\limits } _ { E } f ( x, y, z) dx dy dz = $$

$$ = \ \int\limits _ {E _ {xy} } dx dy \int\limits _ {\phi ( x, y) } ^ { \psi ( x, y) } f ( x, y, z) dz. $$

2) Let the projection of $ E $ on the $ x $- axis be an interval $ [ a, b] $, and let $ E ( x) $ be the intersection of $ E $ with the plane through the point $ x $ parallel to the $ yz $- plane; then

$$ {\int\limits \int\limits \int\limits } _ { E } f ( x, y, z) dx dy dz = \ \int\limits _ { a } ^ { b } dx {\int\limits \int\limits } _ {E ( x) } f ( x, y, z) dy dz. $$

In case $ G $ is a Jordan-measurable domain in the space $ \mathbf R _ {x} ^ {n} $ and $ \phi $ is also continuously differentiable on the closure $ \overline{G}\; $ of $ G $ into $ \mathbf R ^ {n} $, one has the following formula for substitution of variables in the integral of a function $ f $ which is integrable on $ \Gamma = \phi ( G) $:

$$ \tag{1 } \int\limits _ {\phi ( G) } f ( x) dx = \ \int\limits _ { G } f ( \phi ( t)) | J ( t) | dt, $$

where $ J ( t) $ is the Jacobian of the mapping $ \phi $.

The geometrical meaning of the multiple Riemann integral of a function of $ n $ variables is connected with the concept of the $ ( n + 1) $- dimensional Jordan measure $ \mu _ {n + 1 } $: If $ f $ is integrable on a set $ E \subset \mathbf R _ {x} ^ {n} $, $ f ( x) \geq 0 $ on $ E $ and if

$$ A = \{ {( x, y) } : {x \in E, 0 \leq y \leq f ( x) } \} \ \subset \mathbf R _ {xy} ^ {n + 1 } , $$

then

$$ \tag{2 } \int\limits _ { E } f ( x) dx = \mu _ {n + 1 } ( A). $$

A multiple Lebesgue integral is the Lebesgue integral of a function of several variables; the definition is based on the concept of the Lebesgue measure in the $ n $- dimensional Euclidean space. A multiple Lebesgue integral can be reduced to a repeated integral (see Fubini theorem). For continuously-differentiable one-to-one mappings of domains, formula (1) for substitution of variables holds, as well as formula (2), which conveys the geometrical meaning of the multiple Lebesgue integral, with $ \mu _ {n + 1 } $ now being interpreted as the $ ( n + 1) $- dimensional Lebesgue measure.

The concept of a multiple integral carries over to functions integrable on a subset $ A $ of the product $ X \times Y $ of two sets $ X $ and $ Y $, on each of which a $ \sigma $- finite complete non-negative measure, $ \mu _ {x} $ and $ \mu _ {y} $, respectively, has been given; in this situation integration over $ A $ involves the measure $ \mu $ which is the product of $ \mu _ {x} $ and $ \mu _ {y} $.

For functions of several variables one also has a concept of an improper multiple integral (see Improper integral). The concept of a multiple integral is also applied to indefinite integrals of functions of several variables: An indefinite multiple integral is a set function

$$ F ( E) = \int\limits _ { E } f ( x) dx, $$

where $ E $ is a measurable set. For example, if $ f $ is Lebesgue integrable on some set, then it is the symmetric derivative of its indefinite integral $ F ( E) $ almost-everywhere on that set. In this sense (in analogy to the case of functions of one variable), the evaluation of an indefinite integral is the operation inverse to differentiation of set functions.

References

[1]	V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 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]	S.M. Nikol'skii, "A course of mathematical analysis" , 2 , MIR (1977) (Translated from Russian)
[a1]	T.M. Apostol, "Mathematical analysis" , Addison-Wesley (1957)
[a2]	R.G. Bartle, "The elements of real analysis" , Wiley (1976)
[a3]	K.T. Smith, "Primer of modern analysis" , Bogden & Quigley (1971)
[a4]	K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981)