A $5-r$ uniqueness theorem

A Borel-regular Carathéodory outer measure $\Lambda$ on a separable metric space $X$, where $\Lambda$ is invariant with respect to a family $H$ of homeomorphisms from $X$ onto $X$, is unique if $\Lambda$ satisfies a $5$ - $r$ condition at one point in $X$ and if $H$ satisfies Condition I, a condition much weaker than, but related to, the invariance of distance under $H$.