proof of existence of the Lebesgue measure
First, let be the collection of bounded open intervals of the real numbers. As this is a -system (http://planetmath.org/PiSystem), uniqueness of measures extended from a -system (http://planetmath.org/UniquenessOfMeasuresExtendedFromAPiSystem) shows that any measure defined on the -algebra is uniquely determined by its values restricted to . It remains to prove the existence of such a measure.
Define the length of an interval as for . The Lebesgue outer measure is defined as
This is indeed an outer measure (http://planetmath.org/OuterMeasure2) (see construction of outer measures) and, furthermore, for any interval of the form it agrees with the standard definition of length, (see proof that the outer (Lebesgue) measure of an interval is its length (http://planetmath.org/ProofThatTheOuterLebesgueMeasureOfAnIntervalIsItsLength)).
We show that intervals are -measurable (http://planetmath.org/CaratheodorysLemma). Choosing any and interval the definition of gives
So, choosing an arbitrary set and a sequence covering ,
So, from equation (1)
Also, choosing any and using the subadditivity of ,
As is arbitrary, and substituting into (2) shows that
Consequently, intervals of the form are -measurable. As such intervals generate the Borel -algebra and, by Caratheodory’s lemma, the -measurable sets form a -algebra on which is a measure, it follows that the restriction of to the Borel -algebra is itself a measure.
|Title||proof of existence of the Lebesgue measure|
|Date of creation||2013-03-22 18:33:14|
|Last modified on||2013-03-22 18:33:14|
|Last modified by||gel (22282)|