Lebesgue outer measure
Let be a subset of , let be the traditional definition of the length of an interval : If , then . Finally, let be the set consisting of the values
Note that is an outer measure space (http://planetmath.org/OuterMeasure2). In particular:
Lebesgue outer measure is defined for any subset of (and is a -algebra).
for any , and .
Lebesgue outer measure has other nice properties:
The outer measure of an interval is its length: .
is translation-invariant. That is, if we define to be the set , we have for any .
The outer measure satisfies all the axioms of a measure except (countable) additivity. However, it is countably additive when one restricts to at least the Borel sets, as this is the usual construction of Borel measure. This result is roughly contained in the Caratheodory Extension theorem.
|Title||Lebesgue outer measure|
|Date of creation||2013-03-22 11:48:15|
|Last modified on||2013-03-22 11:48:15|
|Last modified by||yark (2760)|