Absolute continuity
Absolute continuity of an integral as a property of the (Lebesgue) integral. Let a function be
-integrable on a set
. The integral of
over
-measurable subsets
is an absolutely continuous set function (see Subsection 3 below) with respect to the measure
if for any
there exists a
such that the integral
for any set
with
. In the general case the integral with respect to a finitely-additive set function with scalar or vectorial
or
is an absolutely continuous function.
A.P. TerekhinV.F. Emel'yanov
Absolute continuity of a measure as a concept in the theory of measures. A measure is absolutely continuous with respect to a measure
if
is an absolutely continuous set function with respect to
. Thus, let
be a finite measure, given together with
on some fixed
-algebra
;
will then be absolutely continuous with respect to
if it follows from
,
, that
. A generalized finite measure (cf. Charge)
is absolutely continuous with respect to a generalized measure
if
, provided that
, where
is the total variation of
.
A.P. Terekhin
Absolute continuity of a function is a stronger notion than continuity. A function defined on a segment
is said to be absolutely continuous if for any
there exists a
such that for any finite system of pairwise non-intersecting intervals
,
, for which
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the inequality
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holds. Any absolutely continuous function on a segment is continuous on this segment. The opposite implication is not true: e.g. the function if
and
is continuous on the segment
, but is not absolutely continuous on it. If, in the definition of an absolutely continuous function, the requirement that the pairwise intersections of intervals
are empty be discarded, then the function will satisfy an even stronger condition: A Lipschitz condition with some constant.
If two functions and
are absolutely continuous, then their sum, difference and product are also absolutely continuous and, if
does not vanish, so is their quotient
. The superposition of two absolutely continuous functions need not be absolutely continuous. However, if the function
is absolutely continuous on a segment
and if
,
, while the function
satisfies a Lipschitz condition on the segment
, then the composite function
is absolutely continuous on
. If a function
, which is absolutely continuous on
, is monotone increasing, while
is absolutely continuous on
, then the function
is also absolutely continuous on
.
An absolutely continuous function maps a set of measure zero into a set of measure zero, and a measurable set into a measurable set. Any continuous function of finite variation which maps each set of measure zero into a set of measure zero is absolutely continuous. Any absolutely continuous function can be represented as the difference of two absolutely continuous non-decreasing functions.
A function that is absolutely continuous on the segment
has a finite variation on this segment and has a finite derivative
at almost every point. The derivative
is summable over this segment, and
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If the derivative of an absolutely continuous function is almost everywhere equal to zero, then the function itself is constant. On the other hand, for any function that is summable on
the function
is absolutely continuous on this segment. Accordingly, the class of functions that are absolutely continuous on a given segment coincides with the class of functions that can be represented as an indefinite Lebesgue integral, i.e. as a Lebesgue integral with a variable upper limit of a certain summable function plus a constant.
If is absolutely continuous on
, then its total variation is
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The concept of absolute continuity can be generalized to include both functions of several variables and set functions (see Subsection 4 below).
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] | V.I. Smirnov, "A course of higher mathematics" , 5 , Addison-Wesley (1964) (Translated from Russian) |
L.D. Kudryavtsev
Absolute continuity of a set function is a concept usually applied to countably-additive functions defined on a -ring
of subsets of a set
. Thus, if
and
are two countably-additive functions defined on
having values in the extended real number line
, then
is absolutely continuous with respect to
(in symbols this is written as
) if
entails
. Here
is the total variation of
:
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and
are measures, known as the positive and negative variations of
; according to the Jordan–Hahn theorem,
. It turns out that the relations 1)
; 2)
,
; 3)
are equivalent. If the measure
is finite,
if and only if for any
there exists a
such that
entails
. According to the Radon–Nikodým theorem, if
are (completely)
-finite, (i.e.
and there exists a sequence
,
such that
![]() |
and if , then there exists on
a finite measurable function
such that
![]() |
Conversely, if is (completely)
-finite and the integral
makes sense, then
as a function of the set
is absolutely continuous with respect to
. If
and
are (completely)
-finite measures on
, there exist uniquely defined (completely)
-finite measures
and
such that
,
and
is singular with respect to
(i.e. there exists a set
such that
,
) (Lebesgue's theorem). A measure, defined on the Borel sets of a finite-dimensional Euclidean space (or, more generally, of a locally compact group), is called absolutely continuous if it is absolutely continuous with respect to the Lebesgue (Haar) measure. A non-negative measure
on the Borel sets of the real line is absolutely continuous if and only if the corresponding distribution function
is absolutely continuous (as a function of a real variable). The concept of absolute continuity of a set function can also be defined for finitely-additive functions and for functions with vector values.
References
[1] | P.R. Halmos, "Measure theory" , v. Nostrand (1950) |
[2] | J. Neveu, "Bases mathématiques du calcul des probabilités" , Masson (1970) |
V.V. Sazonov
Comments
References
[a1] | H.L. Royden, "Real analysis" , Macmillan (1968) |
[a2] | A.C. Zaanen, "Integration" , North-Holland (1967) |
[a3] | W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953) |
[a4] | W. Rudin, "Real and complex analysis" , McGraw-Hill (1966) pp. 98 |
[a5] | A.E. Taylor, "General theory of functions and integration" , Blaisdell (1965) |
[a6] | C.D. Aliprantz, O. Burleinshaw, "Principles of real analysis" , North-Holland (1981) |
Absolute continuity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Absolute_continuity&oldid=15105