I've just had a thought that somebody else has definitely had before but I don't know what they might have called it.
I have a $\sigma$-algebra $(X=\mathbb{R}^n, \Sigma)$ and I want to measure it, but instead of using a map $\Sigma \rightarrow \mathbb{R}$, I want a map $\Sigma \rightarrow (\mathbb{N} \rightarrow \mathbb{R})$ that tells me how much measure a subset has in each dimension, measuring the topological interior separately from the topological boundary. This should give back the regular measure when restricted to the dimension of the space.
I imagine an open ball in 3-space would have measure $(0, 0, 0, \frac{4}{3} \pi r^3, 0...)$, a closed ball would have measure $(0, 0, 4 \pi r^2, \frac{4}{3} \pi r^3, 0...)$, a set of n isolated points would have measure $(n, 0...)$, and so on. I would also hope this can extend to more than 2 dimensions in a single set, so that for example we can have a closed line segment sticking out of a 2-ball and get measure in dimensions 0, 1, and 2.
Can anyone provide pointers to constructions like this in the literature?
e: A less ambiguous way to ask this might be to ask about measures of embeddings of cell complexes in $\mathbb{R}^n$, keeping the measures of embeddings of different dimensional cells in different grades.