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

Famous quotes containing the words statement of the, statement of, statement and/or theorem:

    “It is commonplace that a problem stated is well on its way to solution, for statement of the nature of a problem signifies that the underlying quality is being transformed into determinate distinctions of terms and relations or has become an object of articulate thought.”
    —John Dewey (1859–1952)

    “Eloquence must be grounded on the plainest narrative. Afterwards, it may warm itself until it exhales symbols of every kind and color, speaks only through the most poetic forms; but first and last, it must still be at bottom a biblical statement of fact.”
    —Ralph Waldo Emerson (1803–1882)

    “If we do take statements to be the primary bearers of truth, there seems to be a very simple answer to the question, what is it for them to be true: for a statement to be true is for things to be as they are stated to be.”
    —J.L. (John Langshaw)

    “To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
    —Albert Camus (1913–1960)