Our first result today is that the metric space associated to the measure ring of a measure space 
is complete.
To see this, let 
be a Cauchy sequence in the metric space 
. That is, for every 
there is some 
so that 
for all 
. Unpacking our definitions, each 
must be an element of the measure ring 
with 
, and thus must be (represented by) a measurable subset 
of finite measure. On the side of the distance function, we must have 
for sufficiently large 
and 
.
Let’s recast this in terms of the characteristic functions 
of the sets in our sequence. Indeed, we find that 
, and so
that is, a sequence of sets is Cauchy in 
if and only if its sequence of characteristic functions 
is mean Cauchy. Since mean convergence is complete, the sequence of characteristic functions must converge in mean to some function 
. But mean convergence implies convergence in measure, which is equivalent to a.e. convergence on sets of finite measure, which is what we’re dealing with.
Thus the limiting function must — like the characteristic functions in the sequence — take the value 
or 
almost everywhere. Thus it is (equivalent to) the characteristic function of some set. Since must be measurable — as the limit of a sequence of measurable functions — it’s the characteristic function of a measurable set, which must have finite measure since its measure is the limit of the Cauchy sequence 
. That is, 
, where 
, and 
is the limit of under the metric of . Thus is complete as a metric space.
August 9, 2010
Posted by John Armstrong |
Analysis, Measure Theory |
1 Comment
