# Differentiation of measures

2010 Mathematics Subject Classification: Primary: 28A15 Secondary: 49Q15 [MSN][ZBL]

Some authors use this name for the outcome of the Radon-Nikodym theorem or for the density of the Radon-Nykodim decomposition (see for instance Section 32 of [Ha]).

Other authors use the name for the following theorem which gives an explicit characterization of the Radon-Nykodim decomposition for locally finite Radon measures on the euclidean space. This theorem is used often in Geometric measure theory and credited to Besicovitch.

Theorem (cp. with Theorem 2.12 of [Ma] and Theorem 2 in Section 1.6 of [EG]) Let μμ and νν be two locally finite Radon measures on RnRn. Then,

• the limit

f(x):=limr↓0ν(Br(x))μ(Br(x))f(x):=limr↓0ν(Br(x))μ(Br(x))exists at μμ-a.e. xx and defines a μμ-measurable map;

• the set

S:={x:limr↓0ν(Br(x))μ(Br(x))=∞}(1)(1)S:={x:limr↓0ν(Br(x))μ(Br(x))=∞}is νν-measurable and a μμ-null set;

• νν can be decomposed as νa+νsνa+νs, where

νa(E)=∫Efdμνa(E)=∫Efdμandνs(E)=ν(S∩E).νs(E)=ν(S∩E).Moreover, for μμ-a.e. xx we have:limr↓01μ(Br(x))∫Br(x)|f(y)−f(x)|dμ(y)=0andlimr↓0νs(Br(x))μ(Br(x))=0.(2)(2)limr↓01μ(Br(x))∫Br(x)|f(y)−f(x)|dμ(y)=0andlimr↓0νs(Br(x))μ(Br(x))=0.