Lebesgue measure

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In measure theory, 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 that can be assigned a volume are called Lebesgue measurable; the volume or measure of the Lebesgue measurable set A is denoted by λ(A).

Henri Lebesgue described this measure in the year 1901, followed the next year by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902.[1]

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

Contents

Examples

Properties

The Lebesgue measure on Rn has the following properties:

All the above may be succinctly summarized as follows:

The Lebesgue measure also has the property of being σ-finite.

Null sets

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