Automatic continuity of group operations

We prove that if (G, ·) is a group with a metric, separable, and Baire topology T such that h → g · h is continuous for all g ϵ G and g → g · h is Baire measurable for all h ϵ G, then (G, T) is a topological group. Several consequences of this result concerning free actions of standard Borel groups are established.