# On the convergence of an integral of Hardy's maximal function

Let $$f:\mathbb{R}\times \mathbb{R}^N \to \mathbb{R}^N$$ be an $$L^1$$ function. Assume that $$\mathcal M f(x,y) = \sup_{r< \bar r}\frac{1}{B_r(y)} \int_{B_r(y)} f(x,z)dz \to 0$$ as $$\bar r \to 0$$ for a.e. $$y \in \mathbb{R}^d$$.

Is it true that for a.e. $$y \in \mathbb{R}^d$$ $$\int_I \mathcal M f(x,y)dx \to 0,$$ where $$I \subset \mathbb{R}$$ is an interval?

If not, what counterexample shows that?

At least, if we fix $$R\gg 1$$, does it hold for a.e. $$x \in B_R(0) \setminus E$$, where $$E$$ is a set of arbitrarily small (non zero) measure?