In
measure theory,
Lebesgue
Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
's dominated convergence theorem provides
sufficient conditions under which
almost everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
convergence
Convergence may refer to:
Arts and media Literature
*''Convergence'' (book series), edited by Ruth Nanda Anshen
*Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics:
**A four-part crossover storyline that ...
of a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of
functions implies convergence in the ''L''
1 norm. Its power and utility are two of the primary theoretical advantages of
Lebesgue integration
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Leb ...
over
Riemann integration
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Gö ...
.
In addition to its frequent appearance in mathematical analysis and partial differential equations, it is widely used in
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
, since it gives a sufficient condition for the convergence of
expected values of
random variables.
Statement
Lebesgue's dominated convergence theorem. Let
be a sequence of
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
-valued
measurable functions on a
measure space . Suppose that the sequence
converges pointwise to a function
and is dominated by some integrable function
in the sense that
:
for all numbers ''n'' in the index set of the sequence and all points
.
Then ''f'' is integrable (in the
Lebesgue
Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
sense) and
:
which also implies
:
Remark 1. The statement "''g'' is integrable" means that measurable function
is Lebesgue integrable; i.e.
:
Remark 2. The convergence of the sequence and domination by
can be relaxed to hold only
almost everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
provided the measure space is
complete or
is chosen as a measurable function which agrees everywhere with the everywhere existing pointwise limit. (These precautions are necessary, because otherwise there might exist a
non-measurable subset of a set , hence
might not be measurable.)
Remark 3. If
, the condition that there is a dominating integrable function
can be relaxed to
uniform integrability
In mathematics, uniform integrability is an important concept in real analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties ...
of the sequence (''f
n''), see
Vitali convergence theorem In real analysis and measure theory, the Vitali convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a generalization of the better-known dominated convergence theorem of Henri Lebesgue. It is a characterization of the conv ...
.
Remark 4. While
is Lebesgue integrable, it is not in general
Riemann integrable
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göt ...
. For example, take ''f''
''n'' to be defined in