Integration over higher dimensional closed and bounded subspaces of R^d

[nm19000]

Hi!

I am currently working on an algorithm to compute the average kernel size in Definition 3.8 from this paper https://arxiv.org/pdf/2311.16898

So far I have reformulated the integral to only be over two domains. For the inner integral I would like to approximate an integral of $\int_{\mathcal{C}} f(v) d\lambda(v)$ where the following assumptions are made:

$\mathcal{C}$ is symmetric, closed and bounded, but we only have access to random samples, $f(\cdot)$ is continuous, bounded and positive semi-definite and $\lambda$ is the Lebesgue measure.

In practice the samples from $\mathcal{C}$ are already fixed, ideally no additional assumptions should be placed on the samples.

Is there a way to get an approximation with statistically valid confidence intervals under these assumptions? And ideally also for larger dimensions, such as image data. Otherwise is there research in this direction that could be extended such that one can get this?