Let $G$ be a locally compact Polish (or compact) group acting continuously on a locally compact Polish (or compact) space $X$, and $\mu$ a Borel measure on $X$. To be sure, continuity of the action means that the map $(g, x) \in G \times X \mapsto g \cdot x \in X$ is continuous with respect to the product topology on $G \times X$. Let $X/G$ denote the orbit space endowed with the quotient topology, and $\pi : X \rightarrow X/G$ denote the orbit map. A cross-section to the orbit map is a map $s: X/G \rightarrow X$ satisfying $s \circ \pi = 1_{X}$. If $s$ and $t$ are measurable or continuous cross-sections to the orbit map, then it is known that their images $s(X/G)$ and $t(X/G)$ are measurable (closed in the case of a continuous cross-section with compact $G$ and $X$). What is the relationship between $\mu(s(X/G)$ and $\mu(t(X/G))$? Is it reasonable to expect $\mu(s(X/G)) = \mu(t(X/G))$?
PS: It is enough for me to consider the case of $X = G$, that is, $X$ is the underlying topological space of $G$, and the action of conjugation, and $\mu$ Haar measure on $G$. Thanks.