= Lebesgue decomposition commutes with finite measure suprema
{c}
{title2=$(\bigvee_i\mu_i)^a=\bigvee_i\mu_i^a,\quad(\bigvee_i\mu_i)^s=\bigvee_i\mu_i^s$}
Take <Lebesgue decompositions> relative to a fixed finite positive <measure> $\kappa$. The finite union $S$ of null supports of their singular components is still $\kappa$-null. On $S$ each original <measure> equals its singular component; on $S^c$ it equals its absolutely continuous component. Restriction commutes with the partition formula defining the <finite supremum of positive measures>. Thus the supremum restricts to $\bigvee_i\mu_i^s$ on $S$ and to $\bigvee_i\mu_i^a$ off $S$. The latter is <absolutely continuous with respect to> $\kappa$ because it is bounded by the sum of the absolutely continuous components. Uniqueness of the <Lebesgue decomposition> proves the displayed identities, and the <Radon-Nikodym derivative> of the absolutely continuous component is the pointwise maximum of the individual derivatives.
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