Say I have a countably infinite number of iid random vectors $X\_i:i\in\mathbb{N}$, each uniformly distributed on $[0,1]^k$ with say, $k=2$.


I need to evaluate stuff like:


$E\_{X\_0,X\_1,\ldots}[\int\_{[0,1]^k}\inf\_{i\in\mathbb{N}}\|X\_i-y\|^2dy]$,


which I "believe" is equal to $0$ (this is indeed a random vector quantizer). Regarding my belief, I can at least show that


$E\_{X\_0,X\_1,\ldots,X\_n}[\int\_{[0,1]^k}\min\_{0 \leq i \leq n}\|X\_i-y\|^2dy] \rightarrow 0$ as $n\rightarrow\infty$.


Remark: Obviously writing the first expectation above does not make sense unless one defines how to do it; I do not know how to do it. 


Is there any reference that might be relevant and covers similar problems?



Countably many random vectors and related problems.
1. Yes, you can define properly the first expectation,
see e.g. [PlanetMath](https://planetmath.org/infiniteproductmeasure)
2. Then you have with your notations
$$ 
\mathbb E\_{X\_i : i\in\mathbb N}\int\_{[0,1]^k}\inf\_{i\in\mathbb N}\|X\_i-y\|^2dy\leq \mathbb E\_{X\_i : i\in\mathbb N}\int\_{[0,1]^k}\min\_{1\leq i \leq N}\|X\_i-y\|^2dy = \mathbb E\_{X\_1,\ldots,X\_N}\int\_{[0,1]^k}\min\_{1\leq i \leq N}\|X\_i-y\|^2dy
$$
and use what you know.


Anyway, questions like this should be first asked on math stack exchange, they are not "research level questions".


