Monotone convergence theorem for non-negative numerical measurable functions

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Statement

Let (X,A,μ) be a measure space and let (fn)nNM(A,B(ˉR0))[Note 1] be a sequence of measurable functions, fn:X[0,+]=ˉR0, then[1][2]:

  • if nN[fnfn+1][Note 2] - i.e. (fn)nN is a non-decreasing sequence - then:
    • limn(fn) dμ=limn(fn dμ)[Note 3]

This could be phrased differently; as an alternative statement:

  • Define f:X[0,+] by f:xlimn(fn(x)), this limit exists forall xX as we allow the value +.
    • Then we have:
      1. fM(A,B(ˉR0))[Todo 1] - f is a measurable function itself - and
      2. f dμ=limn(fn dμ)

Proof

Grade: B
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Done on paper? Find it and post here? Do again? Can be found in[1] before claim 9.11 which is (a form of) Fatou's lemma

Notes

  1. Jump up Suppose we had:
    1. fM(A,B(ˉR0)) and
    2. fL+:={f:XˉR | xX[0f(x)]fM(A,B(ˉR))}
    Then these are the same requirements for f, that is M(A,B(ˉR0))=L+
    • I've seen (but not read) a proof and trust the source - Alec -17/April/2017 - 0908
  2. Jump up fg is short for xX[f(x)g(x)]
  3. Jump up Note that for the integral of a non-negative numerical measurable function to be even defined that (as the name suggests) the function must be a measurable function. This is covered in the "alternative statement".

References

  1. Jump up to: 1.0 1.1 Measures, Integrals and Martingales - René L. Schilling
  2. Jump up A Guide To Advanced Real Analysis - Gerald B. Folland

Tasks

  1. Jump up
    TODO: Link to specific proof!