Properties of Metric Spaces of Measure Rings
Today we collect a few facts about the metric space 
associated to a measure ring 
.
First of all, the metric 
on is translation-invariant. That is, if 
, 
, and 
are sets in 
with finite measure, then 
. Indeed, we calculate
Next we need to define an “atom” of a measure ring to be a minimal nonzero element. That is, it’s an element 
so that if 
then either 
or 
. We also define a measure ring to be “non-atomic” if it has no atoms. As a quick result: a totally 
-finite measure ring can have at most countably many atoms, since 
must contain all of them, and no two of them can have a nontrivial intersection — if there were uncountably many of them, any decomposition of could only cover countably many of them.
On the other hand, we define a metric space to be “convex” if for any two distinct elements and there is an element between them. That is, is neither nor , and it satisfies the equation 
. We assert that the metric space of a -finite measure ring is convex if and only if the measure ring is non-atomic.
Let and be elements of . Without loss of generality we can assume that by using the translation-invariance of . Indeed, we can replace with 
and with 
. There will be an element between and if and only if 
is between and 
.
So for any is there an element between and ? Such a will satisfy
This is only possible if 
, which means that 
, and so 
. But for to be between and it cannot be equal to either of them, which means that cannot be an atom for any with 
. Since we can decompose any 
into a countable union of elements of finite measure, no element of infinite measure can be an atom either.
Like this:
Like Loading...
August 12, 2010 -
Posted by John Armstrong |
Analysis, Measure Theory
[...] straightforward to verify that is a homomorphism of boolean rings using the fact that is an atom of [...]
Pingback by Stone’s Representation Theorem I « The Unapologetic Mathematician | August 18, 2010 |
[...] if is a totally finite, non-atomic measure algebra, and if is a dense, decreasing sequence of partitions of , then . Indeed, the [...]
Pingback by Partitions in Measure Algebras « The Unapologetic Mathematician | August 24, 2010 |
[...] say that every separable, non-atomic, normalized measure algebra is isomorphic to the measure ring [...]
Pingback by The Measure Algebra of the Unit Interval « The Unapologetic Mathematician | August 25, 2010 |
Sir,
What else can be said of the metric space (X,d_m) associated to a measure ring (S,m) if we assume more structure for S?
For example I was wondering about this: if S is the Borel sigma algebra on a compact group G with Haar measure, must the associated metric space be compact?
Thank you
I’m not really sure offhand what can be said. Though it’s certainly an interesting question worth investigating.