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]$ ?