Let's say I have a distribution $u\in\mathcal{S}'(\mathbb{R}^n)$ and two function $f\in\mathcal{S}(\mathbb{R}^{n+m})$, $g\in\mathcal{S}(\mathbb{R}^m)$.
Then \begin{align} \tilde{f}(y)=\int_{\mathbb{R}^n}u(x)f(x,y)~dx \end{align} should again be a Schwartz function and thus \begin{align} \int_{\mathbb{R}^m}g(y)\tilde{f}(y)~dy \end{align} is a complex number.
On the other hand, I could first evaluate \begin{align} \tilde{g}(x)=\int_{\mathbb{R}^m}f(x,y)g(y)~dy \end{align} and then $u(\tilde{g})$ is a second complex number.
Lastly, $u\otimes g$ defines a unique distribution in $\mathcal{S}(\mathbb{R}^{n+m})$ by the nuclear/Schwartz kernel theorem and $(u\otimes g)(f)$ is a third complex number.
If $u$ were an integrable function, then it is clear that at least the first two numbers are the same due to Fubini's theorem. Is this still true for all tempered distributions? I imagine one could choose some $(u_n)_{n\in\mathbb{N}}\subset\mathcal{S}(\mathbb{R}^n)$ such that $u_n\rightarrow u$ and since Fubini holds for each term of this sequence it should also hold for the limit. Am I right?
What about the third case?