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With a new proof approach we prove in a more general setting the classicalconvergence theorem that almost everywhere convergence of measurable functionson a finite measure space implies convergence in measure. Specifically, wegeneralize the theorem for the case where the codomain is a separable metricspace and for the case where the limiting map is constant and the codomain isan arbitrary topological space.

Generalization and New Proof for Almost Everywhere Convergence to Imply Local Convergence in Measure