31.3. DIFFERENTIATION OF RADON MEASURES 1095

Also λ µ is given by the formula

λ µ (E)≡∫

EDµ λ (x)dµ

Proof: If x ∈ N, this could happen two ways, either x ∈ Z or Dµ λ (x) fails to exist.It only remains to verify that λ µ given above satisfies λ µ ≪ µ. However, this is obviousbecause if µ (E) = 0, then clearly

∫E Dµ λ (x)dµ = 0.

Since Dµ λ (x) = Dµ λ (x)XNC (x) , it doesn’t matter which we use but maybe Dµ λ (x)doesn’t exist at some points of N.

This is sometimes called the Lebesgue decomposition.