Let $x_0,x_1,\ldots$ be an infinite sequence of real numbers with $x_n \in [0,1],\,\forall n\in\mathbb{N}$. Let $\mathcal{F}_0,\mathcal{F}_1,\ldots,$ be an infinite sequence of disjoint Lebesgue measurable subsets of $[0,1]$ with $\bigcup_{n\in\mathbb{N}}\mathcal{F}_n = [0,1]$. Consider the real number $\delta = \sum_{n\in\mathbb{N}}\int_{\mathcal{F}_n} (x-x_n)^2 \mathrm{d}x$.
So, what I did is the following: Since $\sum_{n\in\mathbb{N}}\int_{\mathcal{F}_n}\mathrm{d}x = 1$, there is an index, say $m$, with $\int_{\mathcal{F}_m}\mathrm{d}x >0$. Now, $\delta \geq \int_{\mathcal{F}_m} (x-x_m)^2 \mathrm{d}x$, and since the integrand is non-zero almost everywhere (except at $x_m$) we have $\delta > 0$.
I guess my derivations are fine. What "worries" me is that I can make $\delta$ arbitrarily small but never $0$. Somehow, and I don't know why, I "feel" that I should be able to take $\{x_0,x_1,\ldots\}$ to be all the rational numbers in $[0,1]$, and get $\delta = 0$ with a "nice" choice of $\mathcal{F}_n$. Am I wrong in my calculations, and if I am right, why is this happening at an intuitive level? Is there any "system" (or a different kind of measure) where I would get $\delta = 0$ (with respect to that measure).