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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 L_1 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 (f_n) 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 f i.e. : \lim_ f_n(x) = f(x) exists for every x \in S. Assume moreover that the sequence f_n is dominated by some integrable function g in the sense that : , f_n(x), \le g(x) for all points x\in S and all n in the index set. Then f_n, f are integrable (in the Lebesgue sense) and :\lim_ \int_S f_n\,d\mu = \int_S \lim_ f_n d\mu = \int_S f\,d\mu. In fact, we have the stronger statement : \lim_ \int_S , f_n-f, \, d\mu = 0. Remark 1. The statement "g is integrable" means that the measurable function g is Lebesgue integrable; i.e since g \ge 0. :\int_S g\,d\mu < \infty. Remark 2. The convergence of the sequence and domination by g can be relaxed to hold only \mu-
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 Z of \mu-measure 0. In fact we can modify the functions f_n (hence its point wise limit f) to be 0 on Z without changing the value of the integrals. (If we insist on e.g. defining f as the limit whenever it exists, we may end up with a non-measurable subset within Z where convergence is violated if the measure space is non complete, and so f might not be measurable. However, there is no harm in ignoring the limit inside the null set Z). We can thus consider the f_n and f as being defined except for a set of \mu-measure 0. Remark 3. If \mu (S) < \infty, the condition that there is a dominating integrable function g can be relaxed to uniform integrability of the sequence (''fn''), 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 f 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 ,1/math>, and let f_n be defined on ,1/math> to take the value 1 on the first n rationals and 0 otherwise. Then f is the
Dirichlet function In mathematics, the Dirichlet function is the indicator function \mathbf_\Q of the set of rational numbers \Q, i.e. \mathbf_\Q(x) = 1 if is a rational number and \mathbf_\Q(x) = 0 if is not a rational number (i.e. is an irrational number). \mathb ...
on ,1/math>, which is not Riemann integrable but is Lebesgue integrable. Remark 5 The stronger version of the dominated convergence theorem can be reformulated as: if a sequence of measurable complex functions f_n is almost everywhere pointwise convergent to a function f and almost everywhere bounded in absolute value by an integrable function then f_n \to f in the
Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
L_1(S, \mu)


Proof

Without loss of generality ''Without loss of generality'' (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as ''without any loss of generality'' or ''with no loss of generality'') is a frequently used expression in mathematics. The term is used to indicat ...
, one can assume that ''f'' is real, because one can split ''f'' into its real and imaginary parts (remember that a sequence of complex numbers converges
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
both its real and imaginary counterparts converge) and apply the
triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of Degeneracy (mathematics)#T ...
at the end. Lebesgue's dominated convergence theorem is a special case of the
Fatou–Lebesgue theorem In mathematics, the Fatou–Lebesgue theorem establishes a chain of inequality (mathematics), inequalities relating the integrals (in the sense of Lebesgue integration, Lebesgue) of the limit superior and limit inferior, limit inferior and the lim ...
. Below, however, is a direct proof that uses Fatou’s lemma as the essential tool. Since ''f'' is the pointwise limit of the sequence (''f''''n'') of measurable functions that are dominated by ''g'', it is also measurable and dominated by ''g'', hence it is integrable. Furthermore, (these will be needed later), : , f-f_n, \le , f, + , f_n, \leq 2g for all ''n'' and : \limsup_ , f-f_n, = 0. The second of these is trivially true (by the very definition of ''f''). Using linearity and monotonicity of the Lebesgue integral, : \left , \int_S - \int_S \= \left, \int_S \\le \int_S. By the reverse Fatou lemma (it is here that we use the fact that , ''f''−''fn'', is bounded above by an integrable function) : \limsup_ \int_S , f-f_n, \,d\mu \le \int_S \limsup_ , f-f_n, \,d\mu = 0, which implies that the limit exists and vanishes i.e. : \lim_ \int_S , f-f_n, \,d\mu= 0. Finally, since : \lim_ \left, \int_S fd\mu-\int_S f_nd\mu\ \leq\lim_ \int_S , f-f_n, \,d\mu= 0. we have that : \lim_ \int_S f_n\,d\mu= \int_S f\,d\mu. The theorem now follows. If the assumptions hold only everywhere, then there exists a set such that the functions ''fn'' 1''S'' \ ''N'' satisfy the assumptions everywhere on ''S''. Then the function ''f''(''x'') defined as the pointwise limit of ''fn''(''x'') for and by for , is measurable and is the pointwise limit of this modified function sequence. The values of these integrals are not influenced by these changes to the integrands on this μ-null set ''N'', so the theorem continues to hold. DCT holds even if ''f''''n'' converges to ''f'' in measure (finite measure) and the dominating function is non-negative almost everywhere.


Discussion of the assumptions

The assumption that the sequence is dominated by some integrable ''g'' cannot be dispensed with. This may be seen as follows: define for ''x'' in the interval and otherwise. Any ''g'' which dominates the sequence must also dominate the pointwise
supremum In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, ...
. Observe that : \int_0^1 h(x)\,dx \ge \int_^1 = \sum_^ \int_ \ge \sum_^ \int_=\sum_^ \frac \to \infty \qquad \textm\to\infty by the divergence of the harmonic series. Hence, the monotonicity of the Lebesgue integral tells us that there exists no integrable function which dominates the sequence on ,1 A direct calculation shows that integration and pointwise limit do not commute for this sequence: : \int_0^1 \lim_ f_n(x)\,dx = 0 \neq 1 = \lim_\int_0^1 f_n(x)\,dx, because the pointwise limit of the sequence is the
zero function 0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...
. Note that the sequence (''fn'') is not even uniformly integrable, hence also the
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 ...
is not applicable.


Bounded convergence theorem

One corollary to the dominated convergence theorem is the bounded convergence theorem, which states that if (''f''''n'') is a sequence of
uniformly bounded In mathematics, a uniformly bounded family of functions is a family of bounded functions that can all be bounded by the same constant. This constant is larger than or equal to the absolute value of any value of any of the functions in the family. ...
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 which converges pointwise on a bounded
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 ...
(i.e. one in which μ(''S'') is finite) to a function ''f'', then the limit ''f'' is an integrable function and :\lim_ \int_S = \int_S. Remark: The pointwise convergence and uniform boundedness of the sequence 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 ''f'' is chosen as a measurable function which agrees μ-almost everywhere with the everywhere existing pointwise limit.


Proof

Since the sequence is uniformly bounded, there is a real number ''M'' such that for all and for all ''n''. Define for all . Then the sequence is dominated by ''g''. Furthermore, ''g'' is integrable since it is a constant function on a set of finite measure. Therefore, the result follows from the dominated convergence theorem. If the assumptions hold only everywhere, then there exists a set such that the functions ''fn''1''S''\''N'' satisfy the assumptions everywhere on ''S''.


Dominated convergence in ''L''''p''-spaces (corollary)

Let (\Omega,\mathcal,\mu) be 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 ...
, a real number and (f_n) a sequence of \mathcal-measurable functions f_n:\Omega\to\Complex\cup\. Assume the sequence (f_n) converges \mu-almost everywhere to an \mathcal-measurable function f, and is dominated by a g \in L^p (cf.
Lp space In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourba ...
), i.e., for every natural number n we have: , f_n, \leq g, μ-almost everywhere. Then all f_n as well as f are in L^p and the sequence (f_n) converges to f in the sense of L^p, i.e.: :\lim_\, f_n-f\, _p =\lim_\left(\int_\Omega , f_n-f, ^p \,d\mu\right)^ = 0. Idea of the proof: Apply the original theorem to the function sequence h_n = , f_n-f, ^p with the dominating function (2g)^p.


Extensions

The dominated convergence theorem applies also to measurable functions with values in a
Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
, with the dominating function still being non-negative and integrable as above. The assumption of convergence almost everywhere can be weakened to require only convergence in measure. The dominated convergence theorem applies also to conditional expectations.Zitkovic 2013, Proposition 10.5.


See also

*
Convergence of random variables In probability theory, there exist several different notions of convergence of sequences of random variables, including ''convergence in probability'', ''convergence in distribution'', and ''almost sure convergence''. The different notions of conve ...
, Convergence in mean *
Monotone convergence theorem In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the good convergence behaviour of monotonic sequences, i.e. sequences that are non- increasing, or non- decreasing. In its ...
(does not require domination by an integrable function but assumes monotonicity of the sequence instead) *
Scheffé's lemma In mathematics, Scheffé's lemma is a proposition in measure theory concerning the convergence of sequences of integrable functions. It states that, if f_n is a sequence of integrable functions on a measure space (X,\Sigma,\mu) that converges alm ...
* Uniform integrability *
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 ...
(a generalization of Lebesgue's dominated convergence theorem)


Notes


References

* * * * * {{Measure theory Theorems in real analysis Theorems in measure theory Theorems in probability theory Articles containing proofs