Measurable Kernels
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 
.
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April 10, 2010 -
Posted by John Armstrong |
Analysis, Measure Theory
[...] There are a bunch of useful facts that we can prove with the help of measurable covers and measurable kernels. These will allow us to move statements we want to show about outer and inner measures to the realm [...]
Pingback by Using Measurable Covers and Kernels I « The Unapologetic Mathematician | April 12, 2010 |
[...] 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. [...]
Pingback by Using Measurable Covers and Kernels II « The Unapologetic Mathematician | April 13, 2010 |