Real Analysis

If  is any subset of , the exterior measure of  where  is any countable covering of E.

A set  of  is measurable if for all  positive there exists and open set  such that .

• 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.

Measure of the union is equal to the sum of the measures when the sets are disjoint. This is not true for exterior measure.

a function  defined on a set E, is measurable if for all  the set  is measurable.

• 1)(finite valued) is measurable if and only if and only if  is measurable for every open set. 
• 2) Continuous functions are measurable. 
• 3) If  is a sequence of measurable functions, all of the following are measurable.