**Properties**

The Lebesgue measure on **R***n* has the following properties:

- If
*A*is a cartesian product of intervals*I*_{1}×*I*_{2}× ... ×*I*_{n}, then*A*is Lebesgue measurable and Here, |*I*| denotes the length of the interval*I*. - 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. - If
*A*is Lebesgue measurable, then so is its complement. - λ(
*A*) ≥ 0 for every Lebesgue measurable set*A*. - 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.) - 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: .)
- If
*A*is an open or closed subset of**R***n*(or even Borel set, see metric space), then*A*is Lebesgue measurable. - 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). - Lebesgue measure is both locally finite and inner regular, and so it is a Radon measure.
- Lebesgue measure is strictly positive on non-empty open sets, and so its support is the whole of
**R***n*. - 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. - If
*A*is Lebesgue measurable and*x*is an element of**R***n*, 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*. - If
*A*is Lebesgue measurable and, then the*dilation of by*defined by is also Lebesgue measurable and has measure - More generally, if
*T*is a linear transformation and*A*is a measurable subset of**R***n*, 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.

Read more about this topic: Lebesgue Measure

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