The glued set function is a Borel measure

For each chart set $q_i=({\phi }_i\circ x_i^{-1})r_{x_i}$. This is nonnegative Borel. The set function ${\nu }_i(E)={\int }_{x_i(E\cap U_i)}{q}_{i}d{\lambda }_n$ is a measure: disjoint Borel sets have disjoint Borel chart images, and the indefinite-integral theorem supplies countable additivity there. In dimension zero it is a singleton weight times its indicator, hence also a measure, even for infinite weight.

For disjoint Borel $E_k$, ${\nu }_i({\bigcup }_k{E}_k)={\sum }_k{\nu }_i(E_k)$; equivalently apply the nonnegative summation theorem to $q_i{\mathbf{1}}_{x_i(E_k\cap U_i)}$. For $a_{ik}={\nu }_i(E_k)\ge 0$, both iterated sums equal ${\sup }_{m,l}{\sum }_{i\le m,k\le l}{a}_{ik}$: a finite selection in any row fits in some finite rectangle, and conversely every rectangle is bounded by either iterated sum. Consequently $\mu ({\bigcup }_k{E}_k)={\sum }_k\mu (E_k)$.

Every ${\nu }_i(\mathrm{\varnothing })=0$, so $\mu (\mathrm{\varnothing })=0$; all values are nonnegative extended reals. Zero coefficients contribute zero, and a one-term family gives its chart measure. Thus the claimed Borel measure exists with no finiteness assumption.