In
measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
,
Lebesgue's dominated convergence theorem gives a mild
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 ...
under which limits and integrals of a sequence of functions can be interchanged. More technically it says that if 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 cal ...
of functions is bounded in absolute value by an integrable function and is
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 ...
pointwise
convergent to a
function then the sequence converges in
to its pointwise limit, and in particular the integral of the limit is the limit of the integrals. 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 L ...
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 or probability calculus 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 expre ...
, since it gives a sufficient condition for the convergence of
expected value
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
s of
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
s.
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 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 in ...
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
i.e.
:
exists for every
. Assume moreover that the sequence
is dominated by some integrable function
in the sense that
:
for all points
and all
in the index set.
Then
are integrable (in the
Lebesgue sense) and
:
.
In fact, we have the stronger statement
:
Remark 1. The statement "
is integrable" means that the measurable function
is Lebesgue integrable; i.e since
.
:
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 ...
i.e. except possibly on a measurable set
of
-measure
. In fact we can modify the functions
(hence its point wise limit
) to be 0 on
without changing the value of the integrals. (If we insist on e.g. defining
as the limit whenever it exists, we may end up with a
non-measurable subset within
where convergence is violated if the measure space is
non complete, and so
might not be measurable. However, there is no harm in ignoring the limit inside the null set
). We can thus consider the
and
as being defined except for a set of
-measure 0.
Remark 3. If
, the condition that there is a dominating integrable function
can be relaxed to
uniform integrability 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 th ...
.
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 Gö ...
. For example, order the rationals in