Measures and integrals

A family of sets $\mathcal{U}$ is said to be downward filtering if to each pair $U_{1}, U_{2}$ in $\mathcal{U}$ there exists a $U_{3}$ in $\mathcal{U}$ with $U_{1} \cap U_{2} \supseteq U_{3}$. A set function $\mu : \mathcal{K}_{0} \rightarrow \mathbb {R}^{+}$ is said to be $\tau$-smooth if $\inf \left\{ \mu K: K \in \mathcal{U}\right\} = \mu ( \cap \mathcal{U})$ for every downward filtering family $\mathcal{U} \subseteq \mathcal{K}_{0}$. Write $\mathcal{K}$ for the $\cap a$-closure of a $(\emptyset , \cup f, \cap f)$ paving $\mathcal{K}_{0}$, the collection of all possible intersections of subclasses of $\mathcal{K}_{0}$.

Say that an increasing functional $T$ on $\mathcal{H}^{+}$ is $\tau$-smooth at zero if $\inf \left\{ T h: h \in \mathcal{V}\right\}$ for each subfamily $\mathcal{V}$ of $\mathcal{H}^{+}$ that is downward filtering to the zero function. (That is, to each $h_{1}$ and $h_{2}$ in $V$ there is an $h_{3}$ in $V$ with $h_{1} \wedge h_{2} \geq h_{3}$ and the pointwise infimum of all functions in $V$ is everywhere zero.) Extend Theorem $\langle 16 \rangle$ to $\tau$-smooth functionals by constructing a $\mathcal{K}$-regular representing measure from the class $\mathcal{K}$ of sets representable as pointwise infima of subclasses of $\mathcal{H}^{+}$.