Is there any way to define a distribution valued integral (in some way analogous to the Bochner integral, e.g.)? The idea comes from the following observation I've heard made in physics and applied math settings, a function $f : \mathbb R \to\mathbb C$ is "just a superposition of deltas", which is just an interesting way of saying: $$f(x) = \int f(y) \delta(x-y)\ dx$$ or, taking $\tau_y$ to be translation by $y$, perhaps more suggestively: $$f(\cdot) = \int f(y) \tau_y \delta(\cdot)\ dy,$$ i.e. $f$ is a superposition of deltas at point $y$ weighted $f(y)dy$. Now one can make similar observations with the integral formula for the Fourier and inverse Fourier transforms, taking the function as a superposition of "plane waves" i.e. complex exponentials. It's worth noting that in the first case the integral isn't really and integral and in the second the integral is only sometimes an integral, though that's not too important at the moment.
So inspired by these observations, is there a nice way to make this line of thinking rigorous, where we are really integrating distributions over a measure. For example, is there a sane way of defining integrals of function $X \to \mathcal D'$ or $X \to \mathcal S'$ where $(X,\mathcal M, \mu)$ is a measure space and we take the functions to be measurable (presumably putting the Borel $\sigma$-algebra on the codomain)?
E.g. the delta function example is of this form, where we define $F : \mathbb R \to \mathcal S'$ by $$F(x) = f(x) \tau_x\delta$$ and the Fourier transform is of this form where we define $F : \mathbb R \to \mathcal{S}$ by $$F(x) = f(x) e^{-2\pi i x \cdot}.$$
One final example is if we gave the $\delta$-mass measure to $\mathbb R$, we should expect that, call it $m$, $$\int \delta_y\ dm(y) = \delta$$ and more generally for any $F : X \to \mathcal S'$, $$\int F(y)\ dm(y) = F(0).$$
The examples we have are quite suggestive and I feel like this intuition should be able to be "cashed out" into something rigorous.