Question

By applying Proposition 5.1 with $s=n=2$, deduce the Lebesgue density theorem $(5,1)$.

   By applying Proposition 5.1 with $s=n=2$, deduce the Lebesgue density theorem $(5,1)$.
Fractal Geometry: Mathematical Foundations and Applications
Fractal Geometry: Mathematical Foundations and Applications
Kenneth Falconer 2nd Edition
Chapter 5, Problem 1 ↓

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1. Proposition 5.1 typically states that for a measurable set \( A \subset \mathbb{R}^n \) and almost every point \( x \in A \), the limit of the ratio of the measure of \( A \) intersected with a ball centered at \( x \) of radius \( r \) to the volume of the ball  Show more…

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By applying Proposition 5.1 with $s=n=2$, deduce the Lebesgue density theorem $(5,1)$.
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