Is it possible to write a definition of integrals using a space-filling curve as opposed to dyadic pavings?

Question

I've recently learned about dyadic pavings in a calculus course and their use in the $n$-dimensional Riemann sum definition. This reminded me of $2$-dimensional space-filling curves which have a similar mesh-like structure as the limit of the fractal goes to infinity. Is it possible to write a similar definition for Riemann sums and integrals of functions from $\Bbb{R}^2$ to $\Bbb{R}$ using some space-filling curve in theory? If so, what would this look like?