Let $G$ be a locally compact group with normalized Haar probability measure $\mu$ and $X$ a locally compact Hausdorff space, $\phi: G \times X \rightarrow X$ a continuous action and $x \in X$. We define $\phi_x: G \rightarrow \bar{x}$ mapping $G$ to the orbit of $x$ by
$$\phi_x(g) = g\cdot x$$
Then there exists a $G$-invariant probability measure $\nu_{\bar{x}}$ on the orbit $\bar{x}$ given by $$\nu_{\bar{x}} = (\phi_x)_*\mu$$ so that for each Borel set $E \in \mathcal{B}(\bar{x}) = \mathcal{B}(X) \cap \bar{x}$,
$$\nu_{\bar{x}}(E) = \mu(\phi_x^{-1}(E))$$
Here $G$-invariant means for each Borel set $E \in \mathcal{B}(X)$ and $g \in G$, $\nu_{\bar{x}}(g \cdot E) = \nu_{\bar{x}}(E)$.
I am wondering when this pushforward probability measure is the only $G$-invariant probability measure on $\bar{x}$. For example, if the orbit is finite, then its unique invariant probability measure is normalized counting measure. Also, if the orthogonal matrices $O(2)$ act on $\mathbb{R}^2$ identified as column matrices by left multiplication, then the orbit of a point is just the circle containing it with unique invariant probability measure given by the normalized arclength measure.
When I looked at a proof of uniqueness of Haar measure, it looked like the proof utilized an integration trick that I don't immediately see working for a group action. Does anyone know if some proof of the uniqueness of Haar measure could be generalized or if there is any reference which discusses this $G$-invariant orbital probability measure and its uniqueness?
Edit:
After thinking about it for a bit. I think if the orbits are compact metric spaces, and $G$ acts isometrically and $G$ is second countable so that it has a countable dense subset $(g_{i})_{i=1}^n$ whose image under $\phi_x$ is then dense in $\bar{x}$.
Then we could make an argument using $\epsilon$-ball covers of closed subsets of $\bar{x}$ to say something like if $\lambda$ and $\rho$ are $G$-invariant measures on $\bar{x}$, then for each $n \in \mathbb{N}$, there exists $\epsilon_n >0$ and $N_n \in \mathbb{N}$ such that $$|1 - N_n \lambda(B(g_1 \cdot x, \epsilon_n))| < 1/n$$ and $$|1 - N_n \rho(B(g_1 \cdot x, \epsilon_n))| < 1/n$$ and then argue that they agree on closed sets by approximation.