Real Analysis

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  1. Define: Exterior Measure
    If Image Upload is any subset of Image Upload, the exterior measure of Image Upload where Image Upload is any countable covering of E.
  2. Define: Measurable set
    A set Image Upload of Image Upload is measurable if for all Image Upload positive there exists and open set Image Upload such that Image Upload.
  3. 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.
  4. 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.
  5. Define: Measurable Function.
    a function Image Upload defined on a set E, is measurable if for all Image Upload the set Image Upload is measurable.
  6. What types of functions are measurable?
    • 1)Image Upload(finite valued) is measurable if and only if and only if Image Upload is measurable for every open set. 
    • 2) Continuous functions are measurable. 
    • 3) If Image Upload is a sequence of measurable functions, all of the following are measurable. Image Upload
Author:
NhanNguyen
ID:
225589
Card Set:
Real Analysis
Updated:
2013-07-02 02:13:28
Tags:
Analysis Math
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Description:
Definitions and Theorems about Measure and Integration.
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