Sets Measurable by an Outer Measure I
An outer measure 
on a hereditary 
-ring 
is nice and all, but it’s not what we really want, which is a measure. In particular, it’s subadditive rather than additive. We want to fix this by restricting to a nice collection of sets within .
Every set 
splits every other set into two pieces: the part that’s in and the part that’s not. What we want to focus on are the sets that split every other set additively. That is, sets 
so that for every set 
we have
We call such sets “-measurable”. Actually, to show that is -measurable we just need to show that 
for every , because the opposite inequality follows from the subadditivity of .
This condition seems sort of contrived at first, and there’s really not much to justify it at first besides the foregoing handwaving. But we will soon see that this definition turns out to be useful. For one thing, the collection 
of -measurable sets is a ring!
The proof of this fact is straightforward, but it feels like pulling a rabbit out of a hat, so follow closely. Given sets -measurable sets and 
, we need to show that their union 
and difference 
are both -measurable as well. Saying that is -measurable means that for every we have
Saying that is -measurable means that for every we have
We can take each of these and plug them into the first equation to find the key equation
Now this key equation works for 
as well as 
. We know that 
and 
, but 
. So, sticking into the key equation we find
But the three terms on the right are the first three terms in the key equation. And so we can replace them and write
which establishes that is -measurable! Behold, the rabbit!
Let’s see if we can do it again. This time, we take 
and stick it into the key equation. We find
Again we can find the three terms on the right of this equation on the right side of the key equation as well. Replacing them in the key equation, we find
which establishes that 
is -measurable as well!
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March 29, 2010 -
Posted by John Armstrong |
Analysis, Measure Theory
That definition is sometimes called “Caratheodory’s condition”.
It’s the approach used by Royden in his Real Analysis. Kind of weird, but powerful.
Indeed, that’s where I first saw it, Zeno. For the moment, though, I’m lifting it from Halmos. Secretly, this is my excuse to read through Halmos, digest, and represent the material (with an algebraist’s eye) in this venue.
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If you want to get rid of the magic trick, you could always just draw a venn diagram. You’re only manipulating 3 sets, so it wouldn’t be terribly cluttered at all. Then the proof would just come down to tagging which of the subsets of the whole union are mu-* measurable
The “magic” is more the way of plugging some equations back into each other in unexpected ways, until the definition we want pops out.
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Thankx