= Hilbert-space construction of dominated measure densities
{c}
{title2=$0\le h=d\nu/d(\mu+\nu)\le1$}
For finite <positive measures> $\mu,\nu$, put $\rho=\mu+\nu$. The <linear functional> $g\mapsto\int g\,d\nu$ is bounded on real $L^2(\rho)$ by the <Cauchy-Schwarz inequality>. The <Riesz representation theorem> gives a measurable $h$ with $\nu(A)=\int_Ah\,d\rho$. Testing indicators of its negative and greater-than-one level sets shows $0\le h\le1$ almost everywhere. Subtraction gives $\mu(A)=\int_A(1-h)\,d\rho$. This constructs the densities without assuming the <Radon-Nikodym theorem> as a prior result.
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