Distribution Theory/Elementary operations

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

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

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

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

$T:\phi \mapsto {\int }_{\mathrm{\Omega }}f(w)T_w(\phi )dw\in {𝒟}^{\prime }(U)$ (resp. $\phi \mapsto {\int }_{\mathrm{\Omega }}f(w)T_w(\phi )dw\in {𝒮}^{\prime }(U)$).

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

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

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

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

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

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

and by the above computation, the net of the $S_{(V_1,\dots ,V_m,x_1,\dots ,x_m)}$ 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. $◻$