Dominated Convergence Theorem
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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 (f_n) 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 f and is dominated by some integrable function g in the sense that : , f_n(x), \le g(x) for all numbers ''n'' in the index set of the sequence and all points x\in S. Then ''f'' is integrable (in the Lebesgue sense) and : \lim_ \int_S , f_n-f, \, d\mu = 0 which also implies :\lim_ \int_S f_n\,d\mu = \int_S f\,d\mu Remark 1. The statement "''g'' is integrable" means that measurable function g is Lebesgue integrable; i.e. :\int_S, g, \,d\mu < \infty. Remark 2. The convergence of the sequence and domination by g 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 f 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 f might not be measurable.) Remark 3. If \mu (S) < \infty, the condition that there is a dominating integrable function g 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 (''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 the ...
. Remark 4. While f is Lebesgue integrable, it is not in general Riemann integrable. For example, take ''f''''n'' to be defined in ,1/math> so that it is one at rational numbers of the form k/m, with and coprime and m>n, and zero everywhere else. The series (''f''''n'') converges pointwise to 0, so ''f'' is identically zero, but , f_n-f, =f_n is not Riemann integrable, since its image in every finite interval is \ and thus the upper and lower
Darboux integral In the branch of mathematics known as real analysis, the Darboux integral is constructed using Darboux sums and is one possible definition of the integral of a function. Darboux integrals are equivalent to Riemann integrals, meaning that a functio ...
s are 1 and 0, respectively.


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" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bi ...
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 degenerate triangles, bu ...
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 inequalities relating the integrals (in the sense of Lebesgue) of the limit inferior and the limit superior of a sequence of functions to the limit inferior and the limit su ...
. 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; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
. 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. Note that the sequence (''fn'') is not even
uniformly integrable 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 th ...
, 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 the ...
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 famil ...
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 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 t ...
, 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 Bourb ...
), 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 (pronounced ) 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 ve ...
, 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 random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to ...
, 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 convergence of monotonic sequences (sequences that are non-decreasing or non-increasing) that are also bounded. Info ...
(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 al ...
*
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 ...
*
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 ...
(a generalization of Lebesgue's dominated convergence theorem)


Notes


References

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