Lebesgue measure

In mathematics, the Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. It is used throughout real analysis, in particular to define Lebesgue integration. Sets which can be assigned a volume are called Lebesgue measurable; the volume or measure of the Lebesgue measurable set A is denoted by λ(A). A Lebesgue measure of ∞ is possible, but even so, assuming the axiom of choice, not all subsets of Rn are Lebesgue measurable. The "strange" behavior of non-measurable sets gives rise to such statements as the Banach-Tarski paradox, a consequence of the axiom of choice.

Related Topics:
Mathematics - Volume - Euclidean space - Real analysis - Lebesgue integration - Axiom of choice - Non-measurable set - Banach-Tarski paradox

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