Proposition (integral of a continuously varying family of distributions against an integrable function with compact essential support is distribution):
Let be a topological space, together with a locally finite measure , where is a -algebra on that contains the Borel -algebra on . Suppose further that has compact essential support, and that
- , where for each , we have (resp. ),
is continuously varying, in the sense that for each (resp. in ) the function is continuous
- (resp. ).
Proof: Define , and let (resp. ) be arbitrary. Let and . Since is locally finite, pick a neighbourhood of such that . Since is continuous, by shrinking if necessary, we may assume that for we have . Since is compact, we may choose so that . Now for each arbitrary finite open cover of and for define the distribution
which is indeed a distribution of the required type ( or . In the particular case of the cover that was constructed above, note that
Note further that tuples of the type , where and is an open cover of , form a directed under the relation
and by the above computation, the net of the converges pointwise to . We conclude since the pointwise limit of continuous linear functions from a barrelled LCTVS into a Hausdorff TVS is continuous and linear.