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Indefinite integral

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An integral

(*)

of a given function of a single variable defined on some interval. It is the collection of all primitives of the given function on this interval. If is defined on an interval of the real axis and is any primitive of it on , that is, for all , then any other primitive of on is of the form , where is a constant. Consequently, the indefinite integral (*) consists of all functions of the form .

The indefinite Lebesgue integral of a summable function on is the collection of all functions of the form

In this case the equality holds, generally speaking, only almost-everywhere on .

An indefinite Lebesgue integral (in the wide sense) of a summable function defined on a measure space with measure is the name for the set function

defined on the collection of all measurable sets in .

References

[1] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
[2] S.M. Nikol'skii, "A course of mathematical analysis" , 1–2 , MIR (1977) (Translated from Russian)
[3] V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian)


Comments

For additional references see Improper integral; Integral.

How to Cite This Entry:
Indefinite integral. L.D. Kudryavtsev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Indefinite_integral&oldid=11527
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098