Dominated convergence theorem
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the L1 norm. 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.
Lebesgue's dominated convergence theorem. Let (fn) be a sequence of complex-valued measurable functions on a measure space (S, Σ, μ). Suppose that the sequence converges pointwise to a function f and is dominated by some integrable function g in the sense that
for all numbers n in the index set of the sequence and all points x ∈ S. Then f is integrable and
which also implies
Remark 1. The statement "g is integrable" means that measurable function g is Lebesgue integrable; i.e.
Remark 2. The convergence of the sequence and domination by g can be relaxed to hold only μ-almost everywhere provided the measure space (S, Σ, μ) is complete or f is chosen as a measurable function which agrees μ-almost everywhere with the μ-almost everywhere existing pointwise limit. (These precautions are necessary, because otherwise there might exist a non-measurable subset of a μ-null set N ∈ Σ, hence f might not be measurable.)
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.
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
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.
we have that
The theorem now follows.
If the assumptions hold only μ-almost everywhere, then there exists a μ-null set N ∈ Σ such that the functions fn 1S \ N satisfy the assumptions everywhere on S. Then the function f(x) defined as the pointwise limit of fn(x) for x ∈ S \ N and by f(x) = 0 for x ∈ N, 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 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 fn(x) = n for x in the interval (0, 1/n] and fn(x) = 0 otherwise. Any g which dominates the sequence must also dominate the pointwise supremum h = supn fn. 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:
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 (S, Σ, μ) (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 μ-almost everywhere, provided the measure space (S, Σ, μ) is complete or f is chosen as a measurable function which agrees μ-almost everywhere with the μ-almost everywhere existing pointwise limit.
Since the sequence is uniformly bounded, there is a real number M such that |fn(x)| ≤ M for all x ∈ S and for all n. Define g(x) = M for all x ∈ S. 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 μ-almost everywhere, then there exists a μ-null set N ∈ Σ such that the functions fn1S\N satisfy the assumptions everywhere on S.
Dominated convergence in Lp-spaces (corollary)
Let be a measure space, 1 ≤ p < ∞ a real number and (fn) a sequence of -measurable functions .
Assume the sequence (fn) converges μ-almost everywhere to an -measurable function f, and is dominated by a (cf. Lp space), i.e., for every natural number n we have: |fn| ≤ g, μ-almost everywhere.
Then all fn as well as f are in and the sequence (fn) converges to f in the sense of , i.e.:
Idea of the proof: Apply the original theorem to the function sequence with the dominating function .
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.
- Convergence of random variables, Convergence in mean
- Monotone convergence theorem (does not require domination by an integrable function but assumes monotonicity of the sequence instead)
- Scheffé's lemma
- Uniform integrability
- Vitali convergence theorem (a generalization of Lebesgue's dominated convergence theorem)
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- Weir, Alan J. (1973). "The Convergence Theorems". Lebesgue Integration and Measure. Cambridge: Cambridge University Press. pp. 93–118. ISBN 0-521-08728-7.
- Williams, D. (1991). Probability with martingales. Cambridge University Press. ISBN 0-521-40605-6.CS1 maint: ref=harv (link)