Lebesgue measure

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, 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.

Properties

The Lebesgue measure has the following properties:

1. If A is a product of intervals of the form I₁ x I₂ x ... x I_n, then A is Lebesgue measurable and λ(A) = |I₁| · ... · |I_n|. Here, |I| denotes the length of the interval I as explained in the article on intervals.
2. If A is a disjoint union of finitely many or countably many disjoint Lebesgue measurable sets, then A is itself Lebesgue measurable and λ(A) is equal to the sum (or infinite series) of the measures of the involved measurable sets.
3. If A is Lebesgue measurable, then so is its complement.
4. λ(A) ≥ 0 for every Lebesgue measurable set A.
5. If A and B are Lebesgue measurable and A is a subset of B, then λ(A) ≤ λ(B). (A consequence of 2, 3 and 4.)
6. Countable unions and intersections of Lebesgue measurable sets are Lebesgue measurable. (A consequence of 2 and 3.)
7. If A is an open or closed subset of Rⁿ (see metric space), then A is Lebesgue measureable.
8. If A is Lebesgue measurable set with λ(A) = 0 (a null set), then every subset of A is also a null set.
9. If A is Lebesgue measurable and x is an element of Rⁿ, then the translation of A by x, defined by A + x = {a + x : a in A}, is also Lebesgue measurable and has the same measure as A.

The Lebesgue measurable sets therefore form a sigma algebra, and λ is a complete translation-invariant measure on that sigma-algebra.

Null sets

A subset of Rⁿ is a null-set if, for every ε > 0, it can be covered with countably many products of n intervals whose total volume is at most ε. All countable sets are null-sets, and so are sets in Rⁿ whose dimension is smaller than n, for instance straight lines or circles in R².

In order to show that a given set A is Lebesgue measurable, one usually tries to find a "nicer" set B which differs from A only by a null set (in the sense that the symmetric difference (A - B) u (B - A) is a null set) and then shows that B can be generated using countable unions and intersections from open or closed sets.

Construction of the Lebesgue measure

For any subset B of Rⁿ, we can define λ^*(B) = inf { vol(M} : M is a countable union of products of intervals, and M contains B }. Here, vol(M) is sum of the product of the lengths of the involved intervals. We then define the set A to be Lebesgue measurable if

λ^*(B) = λ^*(A ∩ B) + λ^*(B - A)

for all sets B. These Lebesgue measurable sets form a sigma algebra, and the Lebesgue measure is defined by λ(A) = λ^*(A).

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Mention connection to Borel measure, [Haar measure]?

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/Talk