 Finite supremum of positive measures

For finite positive measures on the same sigma-algebra, their least common upper bound in the order on measures isEmpty 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 $\epsilon$ 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.