Equivalence of weak and strong modes of measures on topological vector spaces

Modes of a probability measure on a normed space $X$ can be defined by maximising the small-radius limit of the ratio of measures of norm balls. Helin and Burger weakened the definition of such modes by considering only balls whose centres differ by a vector in a topologically dense, proper subspace $E$ of $X$, and posed the question of when these two types of modes coincide. We generalise these definitions, by replacing norm balls with bounded, open neighbourhoods that satisfy a boundary regularity condition, and probability measures with non-atomic measures that are finite on bounded sets. We show that a coincident limiting ratios condition is a necessary and sufficient condition for the equivalence of these two types of modes, and that this condition is satisfied under the assumption that $E$ is dense in $X$.