Let be a pre-measure on an algebra i.e., is a -additive set function on . Then there exists a measure on whose restriction to is . If is another measure on that extends then
with equality .
If is -finite; there exist a sequence of sets such that
with , then is the unique extension of to a measure on .
Proof
Take
with
Then,
Thus
NOTE
Namely,
If , then given , we can choose such that
where . This implies that
Now,
Since was arbitrary,
We can assume
is disjoint in . Take ,
Then
Consider where is the intervals of open sets.
Let and be two disjoint countable dense sets in .
Define and is the counting measure.
NOTE
, Since is dense.
, so is a cover of with respect to .
and are not -finite on . If , then
So, on .
However on because
because are not -finite on .
Borel measure on
Let be increasing and right continuous.
Define
Recall
where interpret as as for .
Define by
Example:-
i) ,
ii)
Lemma
Let be disjoint and
Then
Proof
WLOG, assume . Then
By monotonically, if
By renaming, if necessary, we assume
Since intervals are disjoint and all of them are contained in