Let $f \colon \mathbb{R}^d \to \mathbb{R}_+$. Lebesgue differentiation theorem states that normalized limit of integrals of $f$ by sets shrink to x is equal to $f(x)$. I need analogue of that theorem for the following situation.
Let $S_\varepsilon(t) = \{x \in \mathbb{R}^d \colon t-\varepsilon \leq |x| \leq t+\varepsilon \}$, $\gamma(t) = \{x \in \mathbb{R}^d \colon |x|=t \}$. Then (I guess) when some conditions hold $$ \lim_{\varepsilon \to 0} \frac{1}{|S_\varepsilon(t)|}\int_{S_\varepsilon(t)}f(x)\, dx = \int_{\gamma(t)} f(x) \, dx. $$
Does anyone know such results?