Lebesgue Measure - Properties

Properties

The Lebesgue measure on Rn has the following properties:

1. If A is a cartesian product of intervals I₁ × I₂ × ... × I_n, then A is Lebesgue measurable and Here, |I| denotes the length of the interval I.
2. If A is a disjoint union of 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. (Not a consequence of 2 and 3, because a family of sets that is closed under complements and disjoint countable unions need not be closed under countable unions: .)
7. If A is an open or closed subset of Rn (or even Borel set, see metric space), then A is Lebesgue measurable.
8. If A is a Lebesgue measurable set, then it is "approximately open" and "approximately closed" in the sense of Lebesgue measure (see the regularity theorem for Lebesgue measure).
9. Lebesgue measure is both locally finite and inner regular, and so it is a Radon measure.
10. Lebesgue measure is strictly positive on non-empty open sets, and so its support is the whole of Rn.
11. If A is a Lebesgue measurable set with λ(A) = 0 (a null set), then every subset of A is also a null set. A fortiori, every subset of A is measurable.
12. 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 ∈ A}, is also Lebesgue measurable and has the same measure as A.
13. If A is Lebesgue measurable and, then the dilation of by defined by is also Lebesgue measurable and has measure
14. More generally, if T is a linear transformation and A is a measurable subset of Rn, then T(A) is also Lebesgue measurable and has the measure .

All the above may be succinctly summarized as follows:

The Lebesgue measurable sets form a σ-algebra containing all products of intervals, and λ is the unique complete translation-invariant measure on that σ-algebra with

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