Lebesgue's density theorem states that at almost every point of a measurable set S in En, the metric density of S exists and is 1 and at almost every point of the complement of S, the density of S exists and is 0. This theorem was first proven for E1 by Lebesgue using his theory of integration. It was later proven by Denjoy , Lusin , and Sierpinski  for E1 without the use of integration. The theorem was first proven for En by de la Vallee Poussin.
Proceedings of the Iowa Academy of Science
© Copyright 1958 by the Iowa Academy of Science, Inc.
Martin, N. F. G.
"Exceptional Values of Metric Density,"
Proceedings of the Iowa Academy of Science: Vol. 65:
, Article 49.
Available at: https://scholarworks.uni.edu/pias/vol65/iss1/49