Lebesgue's decomposition theorem

In mathematics, more precisely in measure theory, the Lebesgue decomposition theorem provides a way to decompose a measure into two distinct parts based on their relationship with another measure. == Formal Statement == The theorem states that if ( Ω , Σ ) {\displaystyle (\Omega ,\Sigma )} is a measurable space and μ {\displaystyle \mu } and ν {\displaystyle \nu } are σ-finite signed measures on Σ {\displaystyle \Sigma } , then there exist two uniquely determined σ-finite signed measures ν 0 {\displaystyle \nu _{0}} and ν 1 {\displaystyle \nu _{1}} such that: ν = ν 0 + ν 1 {\displaystyle \nu =\nu _{0}+\nu _{1}\,} ν 0 ≪ μ {\displaystyle \nu _{0}\ll \mu } (that is, ν 0 {\displaystyle \nu _{0}} is absolutely continuous with respect to μ {\displaystyle \mu } ) ν 1 ⊥ μ {\displaystyle \nu _{1}\perp \mu } (that is, ν 1 {\displaystyle \nu _{1}} and μ {\displaystyle \mu } are singular).

Source: Wikipedia — Lebesgue's decomposition theorem (CC BY-SA 4.0)