# Distribution Theory/Elementary operations

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

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

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

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

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

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

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

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

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

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

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

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