Gluing Together Borel Measures

Does anyone know a standard reference for the following, which I assume is true:

X a topological space, $\{U_i\}$ an open cover, $\mu_i$ a collection of regular Borel measures agreeing on overlaps. Then there's a regular Borel measure on X agreeing with the $\mu_i$. If you want, assume X is a manifold.

• On a manifold you could use a partition of unity... – David C. Ullrich Sep 17 '15 at 0:10
• @DavidC.Ullrich Oh good point. And I guess regularity shouldn't be too hard. – Tim kinsella Sep 17 '15 at 0:15
• Come to think of it, I bet you can use a partition of unity on, say, a compact Hausdorff space... I bet. – David C. Ullrich Sep 17 '15 at 4:35
• @DavidC.Ullrich: What if $X$ is not compact? How do you show the finiteness of your sum? – Alex M. Oct 30 '15 at 17:19
• @AlexM. Maybe I'm missing something, but I took David's comment to mean $\mu (f) := \sum_{i=1}^\infty \mu_i(\phi_i f)$, for $\{\phi_i\}$ a partition of unity subordinate to $\{U_i\}$. Does that not work? – Tim kinsella Nov 3 '15 at 19:21