Let $\phi:X\times\mathbb{R}\mapsto X$ a continuous flow where $(X,d)$ is a compact metric space.
I am interested in this question: There is some borel probability measure $\mu$ invariant for $\phi$ such that fades along the flow orbits?, that is, $$\mu(\{ \phi(x,t):t\in \mathbb{R}\})=0\quad \forall x\in X.$$
I was seeing that in the discrete case it is verified with any non-atomic and invariant measure. I just started in the analysis of continuous systems and thought that such observation of the discrete case would be natural in the continuous case, but I find it difficult to find examples. So, in the flows case are there examples not necessarily lebesgue? (Since all examples I can think of are lebesgue)
I appreciate any suggestions.