nonLebesguemeasurábilis

Non-Lebesgue-measurable sets are sets of real numbers that cannot be assigned a "size" or "measure" in the same way that the real numbers, intervals, or more standard subsets of the real line can be. The Lebesgue measure is a generalization of length for intervals, area for rectangles, and volume for solids. It is a powerful tool in analysis, particularly in the theory of integration.

The existence of non-Lebesgue-measurable sets is a consequence of the Axiom of Choice, a fundamental principle in set theory. Without the Axiom of Choice, it can be proven that all subsets of the real line are Lebesgue-measurable. However, with the Axiom of Choice, it is possible to construct sets that defy this measurement.

A common example is the Vitali set. To construct a Vitali set, one considers the real numbers modulo 1 (effectively, the interval [0,1)). We define an equivalence relation where two numbers x and y are equivalent if their difference x-y is a rational number. This partitions the interval [0,1) into disjoint equivalence classes. The Axiom of Choice is used to select exactly one representative from each equivalence class. This collection of representatives forms a Vitali set.

Vitali sets are non-Lebesgue-measurable. If a Vitali set were measurable, its measure would have to be either 0 or a positive number. If the measure is 0, then shifting this set by rational numbers (which are of measure zero) would also have measure 0. The union of countably many sets of measure 0 would also have measure 0. However, by construction, the union of countable shifts of a Vitali set by rational numbers between -1 and 1 covers the interval [0,1) exactly once. This would imply the measure of [0,1) is 0, which contradicts the fact that the Lebesgue measure of [0,1) is 1. If the Vitali set had a positive measure, then by shifting it by rational numbers, we would obtain infinitely many disjoint sets each with the same positive measure, which would lead to an infinite measure for the interval [0,1), also a contradiction. Therefore, Vitali sets cannot be Lebesgue-measurable.