Monotone convergence theorem for non-negative numerical measurable functions
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Contents
[hide]Statement
Let (X,A,μ) be a measure space and let (fn)n∈N⊆M(A,B(ˉR≥0))[Note 1] be a sequence of measurable functions, fn:X→[0,+∞]=ˉR≥0, then[1][2]:
- if ∀n∈N[fn≤fn+1][Note 2] - i.e. (fn)n∈N 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:x↦limn→∞(fn(x)), this limit exists forall x∈X as we allow the value +∞.
- Then we have:
- f∈M(A,B(ˉR≥0))[Todo 1] - f is a measurable function itself - and
- ∫f dμ=limn→∞(∫fn dμ)
- Then we have:
Proof
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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
- Jump up ↑ Suppose we had:
- f∈M(A,B(ˉR≥0)) and
- f∈L+:={f:X→ˉR | ∀x∈X[0≤f(x)]∧f∈M(A,B(ˉR))}
- I've seen (but not read) a proof and trust the source - Alec -17/April/2017 - 0908
- Jump up ↑ f≤g is short for ∀x∈X[f(x)≤g(x)]
- 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
- ↑ Jump up to: 1.0 1.1 Measures, Integrals and Martingales - René L. Schilling
- Jump up ↑ A Guide To Advanced Real Analysis - Gerald B. Folland
Tasks
- Jump up ↑ TODO: Link to specific proof!
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