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In mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a length, area or 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 ?(A). A Lebesgue measure of 8 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.

Lebesgue measure is often denoted , but this should not be confused with the distinct notion of a volume form.

The Lebesgue measure on Rⁿ has the following properties

All the above may be succinctly summarized as follows

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