Proof: Define${\displaystyle A:=\mathrm{esssupp}f}$, and let${\displaystyle \phi \in \mathcal{D}(U)}$ (resp.${\displaystyle \in \mathcal{S}({\mathcal{R}}^n)}$) be arbitrary. Let${\displaystyle x\in A}$ and${\displaystyle ϵ>0}$. Since${\displaystyle \mu }$ is locally finite, pick a neighbourhood${\displaystyle U_x}$ of${\displaystyle x}$ such that. Since${\displaystyle x\mapsto T_x(\phi )}$ is continuous, by shrinking${\displaystyle U_x}$ if necessary, we may assume that for${\displaystyle y\in U}$ we have${\displaystyle |T_x(\phi )-T_y(\phi )|\le ϵ/\mu (U_x)}$. Since${\displaystyle A}$ is compact, we may choose${\displaystyle x_1,\dots ,x_n\in A}$ so that${\displaystyle A=U_{x_1}\cup \cdots \cup U_{x_n}}$. Now for each arbitrary finite open cover${\displaystyle V_1,\dots ,V_m}$ of${\displaystyle A}$ and${\displaystyle x_j\in V_j}$ for${\displaystyle j\in [m]}$ define the distribution

${\displaystyle S_{(V_1,\dots ,V_m,x_1,\dots ,x_m)}(\phi ):=\sum _{j=1}^n{\int }_{V_j\setminus (V_{j-1}\cup \cdots \cup V_1)}f(w)T_{x_j}(\phi )dw}$,

which is indeed a distribution of the required type (${\displaystyle {\mathcal{D}}^{\prime }(U)}$ or${\displaystyle {\mathcal{S}}^{\prime }({\mathbb{R}}^n)}$. In the particular case of the cover that was constructed above, note that

${\displaystyle \left|S_{(U_{x_1},\dots ,U_{x_n},x_1,\dots ,x_n)}(\phi )-T(\phi )\right|\le \sum _{j=1}^n{\int }_{V_j\setminus (V_{j-1}\cup \cdots \cup V_1)}|f(w)||T_w(\phi )-T_{x_j}(\phi )|dw\le ϵ\Vert f{\Vert }_1}$.

Note further that tuples of the type${\displaystyle (V_1,\dots ,V_m,x_1,\dots ,x_m)}$, where${\displaystyle x_j\in V_j}$ and${\displaystyle V_1,\dots ,V_m}$ is an open cover of${\displaystyle A}$, form a directed under the relation

${\displaystyle (V_1,\dots ,V_m,x_1,\dots ,x_m)\le (W_1,\dots ,W_k,y_1,\dots ,y_k):\iff \{x_1,\dots ,x_m\}\subseteq \{y_1,\dots ,y_n\}\wedge \mathrm{\forall }j\in [m]\mathrm{\exists }l\in [k]:W_l\subseteq V_j}$,

and by the above computation, the net of the${\displaystyle S_{(V_1,\dots ,V_m,x_1,\dots ,x_m)}}$ converges pointwise to${\displaystyle T}$. We conclude since thepointwise limit of continuous linear functions from a barrelled LCTVS into a Hausdorff TVS is continuous and linear.${\displaystyle ◻}$

• Book:Distribution Theory