# Convergence in measure

**Convergence in measure** is either of two distinct mathematical concepts both of which generalize
the concept of convergence in probability.

On a finite measure space, both notions are equivalent. Otherwise, convergence in measure can refer to either global convergence in measure or local convergence in measure, depending on the author.

There is a topology, called the **topology of (local) 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

still defines a metric that generates the global convergence in measure.^{[1]}

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.