## 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. $y \in [0,a]$

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