Dominated Convergence Theorem - Statement of The Theorem

Statement of The Theorem

Let {ƒn} be a sequence of real-valued measurable functions on a measure space (S, Σ, μ). Suppose that the sequence converges pointwise to a function ƒ 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 ƒ is integrable and

\lim_{n\to\infty} \int_S |f_n-f|\,d\mu = 0

which also implies

Remarks:

  1. The statement 'g is integrable' is meant in the sense of Lebesgue; that is \int_S|g|\,d\mu < \infty.
  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 ƒ 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 ƒ might not be measurable.)
  3. The condition that there is a dominating integrable function g can be relaxed to uniform integrability of the sequence {ƒn}, see Vitali convergence theorem.

Read more about this topic:  Dominated Convergence Theorem

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