Distribution Theory/Elementary operations

Proposition (integral of a continuously varying family of distributions against an integrable function with compact essential support is distribution):

Let $\Omega$ be a topological space, together with a locally finite measure $\mu :{\mathcal {F}}\to [0,\infty ]$, where ${\mathcal {F}}$ is a $\sigma$-algebra on $\Omega$ that contains the Borel $\sigma$-algebra on $\Omega$. Suppose further that $f\in L^{1}(\Omega ,\mu )$ has compact essential support, and that

$x\mapsto T_{x}$, where for each $x\in \Omega$, we have $T_{x}\in {\mathcal {D}}'(U)$ (resp. $T_{x}\in {\mathcal {S}}'({\mathcal {R}}^{n})$),

is continuously varying, in the sense that for each $\varphi \in {\mathcal {D}}(U)$ (resp. in ${\mathcal {S}}({\mathcal {R}}^{n})$) the function $x\mapsto T_{x}(\varphi )$ is continuous Then also

$T:\varphi \mapsto \int _{\Omega }f(w)T_{w}(\varphi )dw\in {\mathcal {D}}'(U)$ (resp. $\varphi \mapsto \int _{\Omega }f(w)T_{w}(\varphi )dw\in {\mathcal {S}}'(U)$).

Proof: Define $A:=\operatorname {esssupp} f$, and let $\varphi \in {\mathcal {D}}(U)$ (resp. $\in {\mathcal {S}}({\mathcal {R}}^{n})$) be arbitrary. Let $x\in A$ and $\epsilon >0$. Since $\mu$ is locally finite, pick a neighbourhood $U_{x}$ of $x$ such that $\mu (U_{x})<\infty$. Since $x\mapsto T_{x}(\varphi )$ is continuous, by shrinking $U_{x}$ if necessary, we may assume that for $y\in U$ we have $|T_{x}(\varphi )-T_{y}(\varphi )|\leq \epsilon /\mu (U_{x})$. Since $A$ is compact, we may choose $x_{1},\ldots ,x_{n}\in A$ so that $A=U_{x_{1}}\cup \cdots \cup U_{x_{n}}$. Now for each arbitrary finite open cover $V_{1},\ldots ,V_{m}$ of $A$ and $x_{j}\in V_{j}$ for $j\in [m]$ define the distribution

$S_{(V_{1},\ldots ,V_{m},x_{1},\ldots ,x_{m})}(\varphi ):=\sum _{j=1}^{n}\int _{V_{j}\setminus (V_{j-1}\cup \cdots \cup V_{1})}f(w)T_{x_{j}}(\varphi )dw$,

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

$\left|S_{(U_{x_{1}},\ldots ,U_{x_{n}},x_{1},\ldots ,x_{n})}(\varphi )-T(\varphi )\right|\leq \sum _{j=1}^{n}\int _{V_{j}\setminus (V_{j-1}\cup \cdots \cup V_{1})}|f(w)||T_{w}(\varphi )-T_{x_{j}}(\varphi )|dw\leq \epsilon \|f\|_{1}$.

Note further that tuples of the type $(V_{1},\ldots ,V_{m},x_{1},\ldots ,x_{m})$, where $x_{j}\in V_{j}$ and $V_{1},\ldots ,V_{m}$ is an open cover of $A$, form a directed under the relation

$(V_{1},\ldots ,V_{m},x_{1},\ldots ,x_{m})\leq (W_{1},\ldots ,W_{k},y_{1},\ldots ,y_{k}):\Leftrightarrow \{x_{1},\ldots ,x_{m}\}\subseteq \{y_{1},\ldots ,y_{n}\}\wedge \forall j\in [m]\exists l\in [k]:W_{l}\subseteq V_{j}$,

and by the above computation, the net of the $S_{(V_{1},\ldots ,V_{m},x_{1},\ldots ,x_{m})}$ converges pointwise to $T$. We conclude since the pointwise limit of continuous linear functions from a barrelled LCTVS into a Hausdorff TVS is continuous and linear. $\Box$