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?