In
measure theory,
Lebesgue's dominated convergence theorem provides
sufficient condition
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
s 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 t ...
convergence 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 called ...
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 Le ...
over
Riemann integration.
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 o ...
, since it gives a sufficient condition for the convergence of
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
s of
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
s.
Statement
Lebesgue's dominated convergence theorem. Let
be a sequence of
complex-valued
measurable function
In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is i ...
s on a
measure space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
. 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 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 t ...
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
In mathematics, a non-measurable set is a set which cannot be assigned a meaningful "volume". The mathematical existence of such sets is construed to provide information about the notions of length, area and volume in formal set theory. In Ze ...
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, functional analysis and measure theory, and plays a vital role in the theory of martingales.
Measure-theoretic definition
Uniform integrability is an extension to the ...
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 ...
.
Remark 4. While
is Lebesgue integrable, it is not in general
Riemann integrable. For example, take ''f''
''n'' to be defined in