= Finite supremum of positive measures
{title2=$\bigvee_{i=1}^N\mu_i$}
For finite <positive measures> on the same <sigma-algebra>, their least common upper bound in the order on <measures> is
$$
\left(\bigvee_i\mu_i\right)(A)=\sup_{A=\bigsqcup_iA_i}\sum_i\mu_i(A_i).
$$
Empty cells are allowed. This is a <measure>, not merely a set function: for a disjoint <countable union>, intersect any candidate partition with its components for the upper bound, and combine partitions whose errors sum to less than a prescribed $\varepsilon$ for the lower bound. <Countable additivity> of each component gives both inequalities. Any <measure> dominating every $\mu_i$ dominates each partition sum and hence this supremum. The result is bounded above by $\sum_i\mu_i$ and so remains finite.
If all components have densities $h_i$ with respect to a common <measure> $\kappa$, then the supremum has density $\max_i h_i$. For the upper bound integrate the pointwise maximum. For equality, partition by the smallest index attaining that maximum; these are <measurable sets>. For example two <measures> supported on disjoint sets have their sum as supremum, rather than the numerical maximum of their masses on the whole space.
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