Local averages of an integrable function recover the function almost everywhere.Suggest a slogan
(Lebesgue differentiation theorem)
Let \(d\geq1\) and let \(f\colon\RR^d\to\CC\) be Lebesgue integrable. Then for almost every \(x\in\RR^d\), \[ \lim_{\substack{m(B)\to0\\ x\in B}}\frac{1}{m(B)}\int_B f(y)\,dy=f(x), \] where the limit is taken over open balls \(B\subseteq\RR^d\) containing \(x\) as their Lebesgue measure \(m(B)\) tends to \(0\) [@SS05].
Let \(f\colon\RR\to\CC\) be Lebesgue integrable, and for \(h>0\) define \(A_h f\colon\RR\to\CC\) by \[ A_h f(x)\coloneqq\frac{1}{2h}\int_{x-h}^{x+h}f(y)\,dy. \] Then \(A_h f(x)\to f(x)\) as \(h\to0^+\) for almost every \(x\in\RR\), and \(\norm{A_h f-f}_{L^1(\RR)}\to0\) as \(h\to0^+\) [@SS05].