Test occupancy in arbitrary region - Monte Carlo Integration - Uniformity test

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Hello all,
I want to do a Monte Carlo integration (because integral limits make it nontrivial) and of course the sampling method is important in such instances, as visually demonstrated by John D'Errico in a related question:
My own question on the matter is: given an arbitrary region (maybe even 3d or higher, where the visualisation is hard or impossible), is there a metric or a test to verify if the sampling is reasonably uniform over the space sampled? For example if someone takes the code from the related question, is it possible to obtain a global measure of the points distribution uniformity?
An image to think about it: 2 x 10^5 points generated inside an arbitrary alphaShape, I want to check if points are indeed uniformly distributed, but visualisation does not help much here. Searching for more I found this is related to a Complete Spatial Randomness test if I understood correctly.
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