Continuous distribution-valued function induces distribution Suppose that the map $\mathbb{R}^n \to \mathcal{D}'(\mathbb{R}^n), \hspace{3mm}\eta\mapsto E_\eta$ is continuous. Furthermore let $\mu$ be a Radon-measure with compact support. 
I'm having trouble showing that the functional $E$ on $\mathcal{D}(\mathbb{R}^n)$, whose action on a test function $\psi$ is defined by
\begin{equation}
\langle E,\psi\rangle = \int_{\mathbb{R}^n} \langle E_\eta, \psi \rangle d\mu(\eta),
\end{equation}
is in fact a distribution.
 A: The question is a bit ambiguous because the topology on $\mathcal{D}'(\mathbb{R}^n)$ is not specified when mentioning that $\eta\rightarrow E_{\eta}$ is continuous.
I don't see immediately if $E$ is indeed continuous when the chosen topology is the weak-$\ast$ topology. However, if the topology is the strong one (a better choice anyway) then the answer is yes as is shown below.
Firstly, by writing $\mu$ as a difference of positive measures one can reduce the problem to the case where $\mu$ is positive. Moreover, by assumption there is a compact set $K$ on which it is supported so
$$
\langle E,\psi\rangle=\int_K \langle E_\eta,\psi\rangle d\mu(\eta)\ . 
$$
Let $A$ be a bounded set in $\mathcal{D}(\mathbb{R}^n)$ and for a distribution $T$ let us use the notation
$$
||T||_A=\sup_{\psi\in A} |\langle T,\psi\rangle|\ .
$$
These seminorms define the strong topology on $\mathcal{D}'(\mathbb{R}^n)$ and are thus continuous. By composition, the map $K\rightarrow [0,\infty)$, $\eta\mapsto ||E_\eta||_A$ is continuous and therefore, by compactness, is bounded by some constant $M$.
Now let $(\psi_n)$ be a sequence that converges to $\psi$ in $\mathcal{D}(\mathbb{R}^n)$. Let $A=\{\psi_n\ :\ n\in\mathbb{N}\}$ which is bounded.
We have $|\langle E_{\eta},\psi_n\rangle|\le M$ for all $n\in\mathbb{N}$ and $\eta\in K$. Moreover, for fixed $\eta$, $\langle E_\eta,\psi_n\rangle\rightarrow
\langle E_\eta,\psi\rangle$. Therefore, by dominated convergence
$$
\langle E,\psi_n\rangle\rightarrow \langle E,\psi\rangle
$$
and $E$ is a distribution.

Addendum recalling some prerequisites:
Here the space of smooth compactly supported functions $\mathcal{D}(\mathbb{R}^n)$ is a topological vector space. Its topology is defined by an uncountable collection of seminorms which are described explicitly here. As for the notion of bounded set in $\mathcal{D}(\mathbb{R}^n)$, the following are equivalent, for $A\subset\mathcal{D}(\mathbb{R}^n)$:


*

*$A$ is bounded

*For every open set $V\subset\mathcal{D}(\mathbb{R}^n)$ which contains the origin, there exists $\lambda>0$ such that $\lambda A\subset V$.

*For every continuous seminorm $||\cdot||$ on $\mathcal{D}(\mathbb{R}^n)$, we have $\sup_{\psi\in A}||\psi||<\infty$.

*For every seminorm $||\cdot||$ in a defining set of seminorms (e.g., the ones in the MO link I gave above), we have $\sup_{\psi\in A}||\psi||<\infty$.

