Lebesgue outer measure

Lebesgue outer measure is a way to assign a size or "measure" to any subset of real numbers, even those highly irregular or uncountable sets. It involves covering the set with a collection of intervals (like slices of bread) whose total length is as small as possible. The Lebesgue outer measure is the infimum (smallest total length) of all such coverings. This approach generalizes the familiar notion of length to very complicated sets, allowing mathematicians to quantify their size systematically and rigorously, which is essential for advanced calculus and analysis.