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Distribution Theory/Elementary operations

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 Then also

(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.