Card Set Information
Definitions and Theorems about Measure and Integration.
Define: Exterior Measure
is any subset of
, the exterior measure of
is any countable covering of E.
Define: Measurable set
is measurable if for all
positive there exists and open set
What are the 6 properties of Measure ?
1) Every open set is measurable.
2) If the set has exterior measure zero it is measurable.
3) The countable union of measurable sets is measurable.
4) Closed sets are measurable.
5) The complement of a measurable set is measurable.
6) The countable intersection of measurable sets is measurable.
Why is exterior measure insufficient. property does measure have the exterior measure not?
Measure of the union is equal to the sum of the measures when the sets are disjoint. This is not true for exterior measure.
Define: Measurable Function.
defined on a set E, is measurable if for all
What types of functions are measurable?
(finite valued) is measurable if and only if and only if
is measurable for every open set.
2) Continuous functions are measurable.
is a sequence of measurable functions, all of the following are measurable.