Borel -algebra
Let be a topological space. Then the -algebra generated by open sets in is called the Borel -algebra for and is denoted by .
-sets
A set in is called if it is an intersection of open sets.
-sets
A set in is called if it is an union of closed sets.
Exercise
is generated by each of the following class of intervals:
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(Open intervals)
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(Closed intervals)
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(Half-open, left-open intervals)
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(Half-open, right-open intervals)
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(Open lower rays)
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(Open upper rays)
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(Closed upper rays)
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(Closed lower rays)
Proposition
Let be a semi algebra. Then for collection of sets s.t.
if
Proof
Let
It follows that . It was enough to show that is an algebra
Take , Then if if .
Now since s.t.
Therefore:-
Definition: (Measure)
Let be a non empty set equipped with a -algebra .A measure on is function st.
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If is a disjoint sequence of sets in then
Example:-
The pair is called a measurable space.
If is a measure then is a measure space.
Example:-
Let be a countable set and and is a sequence of +ve integers.
Define Check is a measure.
On define
Example
,
Define
Prove that is finitely additive i.e.
then
Now, and
But