Relating group topologies by their continuous points

Abstract:

Let $x$ be a point in a topological group $G$, and for each integer $n$, let $(1/n)x$ be the set $\{ y:ny = x\}$ in $G$. Then I call $x$ a continuous point if for positive integers $n$, the subsets $(1/n)x$ are nonvoid and eventually intersect each neighbourhood of the identity 0. I prove the following result and from it three corollaries. Let $G$ be a divisible abelian group such that $(1/n)0 = \{ 0\}$ for some integer $n \geqslant 2$. Suppose there are two group topologies ${\mathcal {A}_1}$ and ${\mathcal {A}_2}$ defined on $G$ and that $G$ is ${\mathcal {A}_2}$-locally compact and $\sigma$-compact, and define ${\omega _2}$ to be the outer measure derived from the Haar measure ${\mu _2}$ on $(G,{\mathcal {A}_2})$. Also suppose that the ratio of the ${\mathcal {A}_2}$-measure of $\{ nx:x \in A\}$ to the ${\mathcal {A}_2}$-measure of $A$, for any ${\mathcal {A}_2}$-Borel-measurable set $A$ (the ratio is the same for any such $A$ with finite measure), does not exceed 1. Then for each ${\mathcal {A}_2}$-Borel-measurable set $A$ with nonvoid ${\mathcal {A}_1}$-interior, ${\mu _2}(A) \geqslant {\omega _2}({W_1}),{W_1}$ being the subgroup of all points in $G$ which are ${\mathcal {A}_1}$-continuous.