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Concept
Dominated convergence theorem
Summary
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 be a sequence of complex-valued measurable functions on a measure space . Suppose that the sequence converges pointwise to a function and is dominated by some integrable function in the sense that
for all numbers n in the index set of the sequence and all points .
Then f is integrable (in the Lebesgue sense) and
which also implies
Remark 1. The statement "g is integrable" means that measurable function is Lebesgue integrable; i.e.
Remark 2. The convergence of the sequence and domination by can be relaxed to hold only μ-almost everywhere provided the measure space (S, Σ, μ) is complete or 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 might not be measurable.)
Remark 3. If , the condition that there is a dominating integrable function can be relaxed to uniform integrability of the sequence (fn), see Vitali convergence theorem.
Remark 4. While is Lebesgue integrable, it is not in general Riemann integrable. For example, take fn to be defined in so that it is one at rational numbers and zero everywhere else (on the irrationals). The series (fn) converges pointwise to 0, so f is identically zero, but is not Riemann integrable, since its image in every finite interval is and thus the upper and lower Darboux integrals are 1 and 0, respectively.
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