Let $(\mathcal{X},d)$ be a metric space, $\mu$ a locally finite Borel measure of $(\mathcal{X},d)$ and $f$ be a real locally $\mu$-integrable function of $(\mathcal{X},d)$. We say that $x \in \mathcal{X}$ is a Lebesgue point for $(f,\mu)$ if \begin{equation} \frac{1}{\mu(\bar{B_r}(x))}\int_{\bar{B_r}(x)}|f(y)-f(x)|\operatorname{d}\mu(y) \to 0, r\to0^+. \end{equation} If $(\mathcal{X},d)$ is a finite dimensional euclidean space, of course, it is a consequence of Lebesgue-Besicovitch theorem that $\mu$-a.e. point of $\mathcal{X}$ is a Lebesgue point for $(f,\mu)$.
I'm looking for applications of this concept. What I have found so far:
- Fatou's lemma (1906): If $(\mathcal{X},d)$ is the $1$-torus, $\mu$ is its normalized Haar measure, $f \in L^1(\mu)$, and $\mathcal{P}(f)$ is its Poisson transform, then if $x \in \mathcal{X}$ is a Lebesgue point for $(f,\mu)$ then $\mathcal{P}(f)(z) \to f(x), z \to x$ non-tangentially.
- Analogous statement for Cesaro's mean of a Fourier series (I think that the result is due to Lebesgue, but I'm not sure, any reference is appreciated): If $(\mathcal{X},d)$ is the $1$-torus, $\mu$ is its normalized Haar measure, $f \in L^1(\mu)$ then the Cesaro's means of the Fourier series of $f$ converges pointwise to $f(x)$ at each Lebesgue point $x \in \mathcal{X}$ for $(f,\mu)$.
- A general tecnique in the spirit of the two previous results (I found this result in a 1971 book of Stein and Weiss, but are they the first to state it in this form?): If $(\mathcal{X},d)$ is a finite dimensional euclidean space, $\mu$ is its Lebesgue, $f \in L^p(\mu)$, $\varphi $ is a real-valued measurable function of $\mathcal{X}$ dominated by a radial integrable non-increasing function $\psi$, then the $(\varepsilon,\varphi)$-mollified of $f$, i.e. $\varphi_\varepsilon*f$, converges as $\varepsilon \to 0^+$ to $f(x)$ at each Lebesgue point $x \in \mathcal{X}$ of $(f,\mu)$.
Does anyone knows any other application of Lebesgue's point, maybe also w.r.t. more general measure and/or in more general metric spaces that essentially euclidean ones, and/or in different areas of mathematics?