In measure theory, Carathéodory’s lemma is used for constructing measures and, for example, can be applied to the construction of the Lebesgue measure and is used in the proof of Carathéodory’s extension theorem. The idea is that to define a measure on a measurable space we would first construct an outer measure (http://planetmath.org/OuterMeasure2), which is a set function defined on the power set of . Then, this outer measure is restricted to and Carathéodory’s lemma is applied to show that this restriction does in fact result in a measure. For an example of this procedure, see the proof of Carathéodory’s extension theorem.
Given an outer measure on a set , the result first defines a collection of subsets of — the -measurable sets. A subset is called -measurable (or Carathéodory measurable with respect to ) if the equality
holds for every . Then, Caratheodory’s lemma says that a measure is obtained by restricting to the -measurable sets.
Let be an outer measure on a set , and be the class of -measurable sets. Then is a -algebra (http://planetmath.org/SigmaAlgebra) and the restriction of to is a measure.
It should be noted that for any outer measure and sets , subadditivity of implies that the inequality is always satisfied. So, only the reverse inequality is required and consequently is -measurable if and only if
for every .
|Date of creation||2013-03-22 18:33:03|
|Last modified on||2013-03-22 18:33:03|
|Last modified by||gel (22282)|