I need to compute a covariance function of a process, which yields the limit $$I:=\lim_{\varepsilon\to0}\int_{U\times U}dx\,dy\,f(x)\,f(y)\,|B^d(\varepsilon)|^{-1}\mathbb{1}\{||x-y||\leq\varepsilon\} \ ,$$ where $U$ is some domain in $\mathbb{R}^d$, $f\in L^2(U)$, and $|B^d(\varepsilon)|$ is the volume of the $d$-dimensional ball of radius $\varepsilon$.
If I could switch limit with integral, this would be $$I = \int_Udx\,f(x)\ \lim_{\varepsilon\to0}\ |B^d(\varepsilon)|^{-1}\int_{||y-x||<\varepsilon}{dy\,f(y)}$$ and using the Lebesgue differentiation theorem, this last limit of the average of $f$ around $x$ is $f(x)$, hence $I=||f||^2_{L^2}$.
Now, how can I use the Dominated or Monotone convergence theorem to justify that switch of limit and integral? I tried using the Hardy-Littlewood maximal inequality, but I couldn't properly bound the average of $f(x)$.
Note: I checked these Tao's lectures, and in the last inequality of Proposition 1.1 he has a bound that could work for me, if the norm that he calls $||\cdot||_{L^{1,\infty}}$ were actually the $L^\infty$ norm. What does he mean by $L^{1,\infty}$ there?
EDIT: As mentioned in the comments, $L^{1,\infty}$ is a Lorentz space, and I don't think that inequality would work for the DCT or MCT here. Any other thoughts?