Dato un qualsiasi spazio invariante per riordinamenti $X\left(\mathrm{\Omega }\right)$ su un insieme aperto $\mathrm{\Omega }\subset {\mathbb{R}}^n$, si determina il più piccolo spazio invariante per riordinamenti $Y\left(\mathrm{\Omega }\right)$ con la proprietà che se $u:\mathrm{\Omega }\to {\mathbb{R}}^n$ è una applicazione che mantiene l'orientamento e ${\left|Du\right|}^n\in X\left(\mathrm{\Omega }\right)$, allora $detDu$ appartiene localmente a $Y\left(\mathrm{\Omega }\right)$.

Let $X$ be a completely regular Hausdorff space, $E$ a boundedly complete vector lattice, $C_b\left(X\right)$ the space of all, bounded, real-valued continuous functions on $X$, $ℱ$ the algebra generated by the zero-sets of $X$, and $\mu C_b\left(X\right)\to E$ a positive linear map. First we give a new proof that $\mu$ extends to a unique, finitely additive measure $\mu ℱ\to E^{+}$ such that $\nu$ is inner regular by zero-sets and outer regular by cozero sets. Then some order-convergence theorems about nets of $E^{+}$-valued finitely additive measures on $ℱ$ are proved, which extend...