The Elements of Integration and Lebesgue Measure (Wiley Classics Library) by Robert G. Bartle

Author:Robert G. Bartle [Bartle, Robert G.]
Language: eng
Format: azw3
Publisher: Wiley
Published: 2014-08-20T16:00:00+00:00

for all E in X. Let A = {x : g(x) = 0}, and let B = {x :g(x)> 0}, so that A ∩ B = ∅, and X = A ∪ B.

Define λ1 and λ2 for E in X by

Since µ(A) = 0, it follows that λ1 ⊥ µ. To see that λ2 µ., observe that if µ(E) = 0, then

so that g(x) = 0 for v-almost all x in E. Hence v(E ∩ B) = 0; since A v,

Clearly λ = λ1 + λ2, so the existence of this decomposition is affirmed.

To establish the uniqueness of the decomposition, use the observation that if α is a measure such that α µ, and α ⊥ µ, then α = 0.

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