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Wikipedia / Dominated convergence theorem

In measure theory, Lebesgue's dominated convergence theorem gives a mild sufficient condition under which limits and integrals of a sequence of functions can be interchanged. More technically it says that if a sequence of functions is bounded in absolute value by an integrable function and is almost everywhere pointwise convergent to a function 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 over Riemann integration.

In addition to its frequent appearance in mathematical analysis and partial differential equations, it is widely used in probability theory, since it gives a sufficient condition for the convergence of expected values of random variables.

Statement

Lebesgue's dominated convergence theorem. be a sequence of complex-valued measurable functions on a measure space . Suppose that the sequence converges pointwise i.e.

in the sense that

are integrable (in the Lebesgue sense) and

  1. .

In fact, we have the stronger statement

Remark 1..

Remark 2.-almost everywhere as the limit whenever it exists, we may end up with a non-measurable subset where convergence is violated if the measure space is non complete-measure 0.

Remark 3. can be relaxed to uniform integrability of the sequence (fn), see Vitali convergence theorem.

Remark 4. is Lebesgue integrable, it is not in general Riemann integrable is the Dirichlet function, which is not Riemann integrable but is Lebesgue integrable.

Remark 5 in the Banach space

Proof

Without loss of generality, 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 both its real and imaginary counterparts converge) and apply the triangle inequality at the end.

Lebesgue's dominated convergence theorem is a special case of the Fatou–Lebesgue theorem. Below, however, is a direct proof that uses Fatou’s lemma as the essential tool.

Since f is the pointwise limit of the sequence (fn) 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),

for all n and

is the limit superior.

The second of these is trivially true (by the very definition of f). Using linearity and monotonicity of the Lebesgue integral,

By the reverse Fatou lemma (it is here that we use the fact that |f−fn| is bounded above by an integrable function)

which implies that the limit exists and vanishes i.e.

Finally, since

we have that

The theorem now follows.

If the assumptions hold only everywhere, then there exists a set such that the functions fn 1S \ N satisfy the assumptions everywhere onS. 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 setN, so the theorem continues to hold.

DCT holds even if fn 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 . Observe that

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 [0,1]. A direct calculation shows that integration and pointwise limit do not commute for this sequence:

because the pointwise limit of the sequence is the zero function. Note that the sequence (fn) is not even uniformly integrable, hence also the Vitali convergence theorem is not applicable.

Bounded convergence theorem

One corollary to the dominated convergence theorem is the bounded convergence theorem, which states that if (fn) is a sequence of uniformly bounded complex-valued measurable functions which converges pointwise on a bounded measure space (i.e. one in which μ(S) is finite) to a function f, then the limit f is an integrable function and

Remark: The pointwise convergence and uniform boundedness of the sequence can be relaxed to hold only almost everywhere, 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 fn1S\N satisfy the assumptions everywhere onS.

Dominated convergence in Lp-spaces (corollary)

be a measure space, .

(cf. Lp space, μ-almost everywhere.

in L^p, i.e.:

.

Extensions

The dominated convergence theorem applies also to measurable functions with values in a Banach space, 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.

See also

Notes

References