Caption Copyright-free illustration of Lebesgue’s Density Theorem applied to a square. Description This diagram visualizes Lebesgue’s Density Theorem for a measurable set in R2\mathbb{R}^2R2. The blue square represents the set.

A point in the interior has density = 1, meaning almost all of its shrinking neighborhoods lie inside the set. Points on the edges have density = 1/2, as half of each small neighborhood lies inside the square. Points on the corners have density = 1/4, since only a quarter of each neighborhood is inside the set. Points outside the square have density = 0, as neighborhoods contain no part of the set.

Arrows indicate where densities of 1/21/21/2 and 1/41/41/4 occur. This image is copyright-free and suitable for educational and Wikimedia use.

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