RadonNikodym derivatives. (i) If and are -finite measures over (X , A) and is

Radon–Nikodym derivatives.

(i) If λ and μ are σ-finite measures over (X , A) and μ is absolutely continuous with respect to λ, then



f dμ =



f dμ

dλ

dλ

for any μ-integrable function f .

(ii) If λ, μ, and ν are σ-finite measures over (X , A) such that ν is absolutely continuous with respect to μ and μ with respect to λ, then dν

dλ = dν

dμ

dμ

dλ

a.e. λ.

(iii) If μ and ν are σ-finite measures„ which are equivalent in the sense that each is absolutely continuous with respect to the other, then dν

dμ =

dμ

dν

!−1 a.e. μ, ν.

(iv) If
μk , k = 1, 2,..., and μ are finite measures over (X , A) such that ∞
k=1 μk (A) = μ(A) for all A ∈ A, and if the μk are absolutely continuous with respect to a σ-finite measure λ, then μ is absolutely continuous with respect to λ, and d n k=1 μk dλ = n k=1 dμk dλ , lim n→∞
d n k=1 μk dλ = dμ
dλ
a.e. λ.
[(i): The equation in question holds when f is the indicator of a set, hence when f is simple, and therefore for all integrable f .
(ii): Apply (i) with f = dν/dμ.]