When $\lim_{r \to 0}\frac{1}{|B_r(\bar x)|}\int_0^a\int_{B_r(\bar x)} u(y,x) dx dy = 0$ implies $u(y,\bar x)=0$ for a.e. $y \in [0,a]$

Question

Let $f$ be a nonnegative $L^\infty([0,a], L^1(\mathbb{R}^N))$ function. Fix any $\bar x \in \mathbb{R}^N$ where $f$ is defined. If $$\lim_{r \to 0}\frac{1}{|B_r(\bar x)|}\int_0^a\int_{B_r(\bar x)} f(y,x) dx dy = 0,$$ does this imply $$f(y,\bar x)=0$$ for a.e. $y \in [0,a]$?

Answer 1

No. Simply let $f(y, \overline{x}) = 1$ for all $y$, and $f(y, x) = 0$ for all $x, y$ where $x \neq \overline{x}$.