Using Measurable Covers and Kernels II
Following yesterday’s post, here are some more useful facts that we can prove with the help of measurable covers and measurable kernels.
If 
and 
are disjoint sets in 
, then
Here, we take 
to be a measurable cover of and 
to be a measurable kernel of 
. The difference 
must be contained in , and so
On the other hand, we can take to be a measurable kernel of and to be a measurable cover of . Now the difference is contained in , and we find
Now, if 
, then for every 
whatsoever, we have
We can take 
and 
and stick them into the previous result to find
But since , we know that 
, and this establishes our result.
Interestingly, we can use this method of inner measures as an alternative approach to our extension theorems. If 
is a 
-finite measure on a ring 
, and if 
is the induced outer measure on 
, then for every set 
of finite measure and every 
we have
Then if and are two sets in such that 
, then we find
and so we can use this formula as the definition of the inner measure 
. Then we can define a set 
with 
to be -measurable if the inner and outer measures match: 
. And from here, the rest of the theory is as before.
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April 13, 2010 -
Posted by John Armstrong |
Analysis, Measure Theory
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