Let $f \in L^1(\mathbb{R}^n)$ and denote by $B(x,r)$ the ball of radius $r$ centered at $x$. The Lebesgue differentiation theorem asserts that the average of $f$ over $B(x,r)$ converges to $f(x)$ for almost every $x \in \mathbb{R}^n$ as $r \to 0$. The theory of differentiation of integrals aims to generalize this result by replacing balls with other bounded sets. For instance, can we use rectangles instead of balls? In this talk, I will present some classical results, the main tools involved in this theory, and some open questions.

I will also talk about the almost everywhere convergence of two-parameter ergodic averages over rectangles in the plane. There are indeed several analogies between these two topics, and I will try to highlight these connections.