Let $(X,\mathfrak{M},\mu)$ be a positive finite measure space, then
define $\rho(A,B)=\int\_X |\chi\_A-\chi\_B|d\mu$.


Is $(\mathfrak{M},\rho)$ a complete metric space(modulo sets of measure 0)?


I am trying very hard to look for any references, but I cannot find any.


So, if $(\mathfrak{M},\rho)$ is a complete metric space, how can I prove that? Actually, the completeness of this metric space is my major problem.



Is this metric space complete?
To complement Michael Renardy's response, a reference is: "1.12.6. Theorem" in Bogachev's Measure Theory, Volume 1. The proof goes as follows. Take any Cauchy sequence in the metric, then pass to a subsequence in which the mutual distances converge to zero very fast, then the original sequence converges in the metric to the set-theoretic limsup of the subsequence.


