Completeness of the Metric Space of a Measure Space
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.