The Radon-Nikodym Theorem

More documents

Recommendations

Info

A measurable set A is said to be positive(resp. negative) with respect to a signedmeasure if ν(E) ≥ 0 (resp. ν(E) ≤ 0) for everymeasurable set E⊆A.Hahn Decomposition Theorem:Let ν be a signed measure on a measurablespace (X,M). Then there is a positive set A and anegative set B such that X = A⋃B and A⋂B = ∅.
The Radon-Nikodym TheoremIf (X,M) is a measurable space and μ and ν aremeasures on M, we say that ν is absolutelycontinuous with respect to μ, in symbols ν≪μ, ifν(E) = 0 for every measurable set E for whichμ(E) = 0.
Page 1: The Radon-NikodymTheoremν(E) = ∫
Page 6 and 7: Theorem (J. Radon - 1913, O. Nikody
Page 8 and 9: Proof of the theorem:Let C be the c
Page 10 and 11: Set E 1 = A 1 , E 2 =A 2 -A 1 ,...,
Page 12 and 13: Let f 0 : X ➝ [0,+∞] be the fun
Page 14 and 15: From (1) and (2) we have α = ∫f
Page 16 and 17: Suppose ν 0 ≠ 0. By the previous
Page 18 and 19: Then g ∈ C and hence ∫ g dμ
Page 20 and 21: Example: One cannot drop the assump
Page 22 and 23: Then there exists c ∈ X such that
Page 24 and 25: Theorem (Radon-Nikodym for signed m

The Radon-Nikodym Theorem

Create successful ePaper yourself

Delete template?

Save as template?