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Prove that if A is a set of positive, finite Lebesgue measure, then almost every point of A is a density point.

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A point x of a measurable subset A of the reals is called a density point if

m( A intersection [x-h, x+h] ) / 2h goes to 1 as h goes to 0

where m is the Lebesgue measure.

Prove that if A is a set of positive, finite Lebesgue measure, then almost every point of A is a density point.

I would like to note that I can use any results from Royden's Real Analysis book.

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This solution is comprised of a detailed explanation to prove that if A is a set of positive, finite Lebesgue measure, then almost every point of A is a density point.

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