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Let

- $G\subseteq\mathbb R^d$ and $$\mathcal D:=C_c^\infty(G)$$ be equipped with some topology $\tau$
- $\mathcal D’$ be the dual space of $\mathcal D$ and $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the dual pairing between $\mathcal D’$ and $\mathcal D$
- $\xi$ be a $\mathcal D’$-valued random variable

How can we show, that $$\operatorname E[\xi]:\mathcal D\to\mathbb R\;,\;\;\;\varphi\mapsto\operatorname E\left[\langle\xi,\varphi\rangle\right]\tag 1$$ is an element of $\mathcal D’$?

Clearly, the mapping $(1)$ is linear. But how can we show, that it is continuous and that $\operatorname E\left[\langle\xi,\varphi\rangle\right]$ is finite?

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Maybe the answer depends on the choice of $\tau$. Maybe $\tau$ must be chosen in a way such that each linear and continuous mapping $\mathcal D\to\mathbb R$ is bounded.

In the Wikipedia article about distributions they equip $\mathcal D$ with a locally convex topology. Do we need this specific topology? Or can we use an arbitrary topology?

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There is a “standard” locally convex topology on $\mathcal{D}$ generated by semi-norms (see the answer to your other question), and pretty much every user of generalized stochastic processes uses it (e.g. Gelfand & Vilenkin). It’s always better to choose your topology *before* going on to work with continuity, dual space, etc. Local convexity is about the weakest assumption you would want to make, but unless you’re driving to prove the most general result possible (not likely), just assume the standard topology. That being said, bounded linear functionals (i.e. $T:\mathcal{D}\rightarrow\Bbb{R}$) are *always* continuous; see Rudin (FA) Thm. 1.18.

As for showing that the mean of $\xi$ is an element of $\mathcal{D}^\prime$, you’re on the right track conceptually, but you really need to be more specific about the process $\xi$ to make concrete statements. Typically one *assumes* things like “$\xi$ is a second-order process” which means that it has a well-defined mean $E[\xi]$ and correlation functional $B(\varphi,\psi) = E[\xi(\varphi)\xi(\psi)]$.

To explain a bit more, recall that for each fixed $\varphi\in\mathcal{D}$, $\langle \xi,\varphi\rangle$ is an $\Bbb{R}$-valued random variable with distribution $p_\varphi(x)$. This distribution might or might not have a well-defined expectation, so $E[\langle \xi,\varphi\rangle]$ might or might not be finite – this has everything to do with how the process $\xi$ behaves. For instance, if $\xi$ is Gaussian (a common but not necessary assumption), then $p_\varphi$ is Gaussian and hence has a finite first moment, etc.

As for continuity, I would prove boundedness instead. That is, show that $\varphi\mapsto E[\langle \xi,\varphi\rangle]$ is a bounded map from $\mathcal{D}$ to $\Bbb{R}$. Again, this requires assumptions on the process $\xi$!

Just out of curiosity, what source are you working from? I would highly recommend reading through Gelfand and Vilenkin “Generalized Functions Vol. 4”.

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