construction of outer measures
The following theorem is used in measure theory to construct outer measures (http://planetmath.org/OuterMeasure2) on a set , starting with a non-negative function on a collection of subsets of . For example, if we take to be the real numbers, to be the collection of bounded open intervals of and define by for real numbers , then the Lebesgue outer measure is obtained.
Let be a set, be a family of subsets of containing the empty set and be a function satisfying . Then the function defined by
is an outer measure.
As , equation (1) defining gives
As is arbitrary, this proves subadditivity (2). ∎
Although this result is rather general, placing few restrictions on the function , there is no guarantee that the outer measure will agree with for the sets in nor that will consist of -measurable (http://planetmath.org/CaratheodorysLemma) sets.
For example, if , consists of the bounded open intervals, and for real numbers , then .
Alternatively if for all then it follows that so
and is not -measurable.
|Title||construction of outer measures|
|Date of creation||2013-03-22 18:33:17|
|Last modified on||2013-03-22 18:33:17|
|Last modified by||gel (22282)|