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.