 High-density interval in a positive-measure subset of the real line

ID: high-density-interval-in-a-positive-measure-subset-of-the-real-line

If a measurable set

E \subseteq R

has positive measure, then for every

ε > 0

there is a bounded interval

I

such that

m (E \cap I) > (1 - ε) m (I) .

Choose a density-one point of a finite positive-measure portion of

E

using the Lebesgue density theorem, and take a sufficiently small interval centred there.

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