Bad is null

We show that on any $C^2$ manifold, almost every point is not badly approximable. More generally, we develop a broad framework for studying badly approximable points in a metric space $X$ equipped with a $\sigma$-finite doubling Borel regular measure $\mu$. We establish that under mild assumptions, the $\mu$-measure of the set of badly approximable points is always zero. The result is achieved by introducing and establishing a `constant invariance' property for a large class of limsup sets of neighbourhoods of subsets of a metric measure space.