= Solution
For the finite nonnegative <measures> there exist a unique nonnegative integrable density $h\in L^1(\kappa)$, up to $\kappa$-almost-everywhere equality, and a unique nonnegative <measure> $\mu^s\perp\kappa$, such that
$$
\boxed{\mu(A)=\int_Ah\,d\kappa+\mu^s(A)\quad(A\in\mathfrak M).}
$$
This combines the <Lebesgue decomposition theorem> with the <Radon-Nikodym theorem>. In particular $\mu^a=h\kappa$ and $h=d\mu^a/d\kappa$; if $\mu\ll\kappa$ the singular term is zero.
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