Lebesgue decomposition from a sum-measure density
= Lebesgue decomposition from a sum-measure density
{c}
{title2=$B=\{h=1\},\quad f=h/(1-h)\text{ on }B^c$}
Use the <Hilbert-space construction of dominated measure densities> with $\rho=\mu+\nu$. The set $B=\{h=1\}$ is a <null set> for $\mu$. Put $f=0$ on $B$ and $f=h/(1-h)$ elsewhere. Then $\int f\,d\mu=\nu(B^c)<\infty$ and $\nu(A)=\int_Af\,d\mu+\nu(A\cap B)$. This is the <Lebesgue decomposition> into a density part and a part concentrated on a $\mu$-<null set>.