Two measures \(ho, \sigma\) defined on the same measurable space \((X, \mathscr{A})\) are singular, if there is a set \(N \in \mathscr{A}\) such that \(ho(N)=\sigma\left(N^{c}ight)=0\). If this is the case, we write \(ho \perp \sigma\). Steps (i)-(iv) below show the so-called Lebesgue decomposition theorem: if \(\mu, u\) are \(\sigma\)-finite measures, then there is a decomposition (unique up to null sets) \(u=u^{\circ}+u^{\perp}\), where \(u^{\circ} \ll \mu\) and \(u^{\perp} \perp \mu\).

(i) \(u \ll u+\mu\) and there is a density \(f=d u / d(u+\mu)\).

(ii) The density satisfies \(0 \leqslant f \leqslant 1\) and \((1-f) u=f \mu\).

(iii) \(u^{\circ}(A):=u(A \cap\{f<1\})\) and \(u^{\perp}(A)=u(A \cap\{f=1\})\).

(iv) Uniqueness follows from \(d u^{\circ} / d \mu=f /(1-f) \mathbb{1}_{\{f<1\}}\).