Throughout, f and fn are measurable functions X → R.
Global convergence in measure implies local convergence in measure. The converse, however, is false; i.e., local convergence in measure is strictly weaker than global convergence in measure, in general.
If, however, or, more generally, if f and all the fn vanish outside some set of finite measure, then the distinction between local and global convergence in measure disappears.
If μ is σ-finite and converges to f in measure, there is a subsequence converging to falmost everywhere. The assumption of σ-finiteness is not necessary in the case of global convergence in measure.
If μ is σ-finite, converges to f locally in measure if and only if every subsequence has in turn a subsequence that converges to f almost everywhere.
In particular, if converges to f almost everywhere, then converges to f locally in measure. The converse is false.
If f and fn are in Lp for some p > 0 and converges to f in the p-norm, then converges to f globally in measure. The converse is false.
If fn converges to f in measure and gn converges to g in measure then fn + gn converges to f + g in measure. Additionally, if the measure space is finite, fngn also converges to fg.
The sequence converges to f locally in measure, but does not converge to f globally in measure.
The sequence where and
converges to 0 globally in measure; but for no x does fn converge to zero. Hence fails to converge to f almost everywhere.
The sequence converges to f almost everywhere and globally in measure, but not in the p-norm for any.
Topology
There is a topology, called the topology of convergence in measure, on the collection of measurable functions from X such that local convergence in measure corresponds to convergence on that topology. This topology is defined by the family of pseudometrics where In general, one may restrict oneself to some subfamily of sets F. It suffices that for each of finite measure and there existsF in the family such that When, we may consider only one metric, so the topology of convergence in finite measure is metrizable. If is an arbitrary measure finite or not, then still defines a metric that generates the global convergence in measure. Because this topology is generated by a family of pseudometrics, it is uniformizable. Working with uniform structures instead of topologies allows us to formulate uniform properties such as Cauchyness.
