A measurable kernel is the flip side of a measurable cover. Specifically, given 
, a measurable kernel of 
is a set 
such that 
, and if for every 
with 
we have 
. And, as it happens, every set has a measurable kernel.
To find it, let 
be a measurable cover of . Then let 
be a measurable cover of 
, and set 
. Since contains , we find
.
If , then
Since was picked to be a measurable cover of , we conclude that , as we hoped.
Now if 
is a measurable kernel of , then 
. Indeed, since , we have 
. If this inequality is strict then 
, and there must be some 
with 
and 
. But 
, while 
, contradicting the fact that was chosen to be a measurable kernel of .
The symmetric difference of any two measurable kernels is negligible. Given two measurable kernels 
and 
, we know that 
. This implies that 
, and thus 
. Similarly, 
, and thus 
.
April 10, 2010
Posted by John Armstrong |
Analysis, Measure Theory |
2 Comments
