So apparently the [Krylov-Bogoliubov theorem](https://en.wikipedia.org/wiki/Krylov%E2%80%93Bogolyubov_theorem) says that every continuous function $f:X\to X$ on a compact metrizable space $X$ has an invariant probability measure $\mu$.


Of course, if $X$ is just a single point then there's only one such $\mu$. Also, if $f$ is constant.


So assume $X$ is the Cantor space $\{0,1\}^{\mathbb N}$, or the unit interval $[0,1]$, and $f$ is invertible.



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> Must there be uncountably many such $\mu$? Must there be an uncountable family of mutually singular $\mu$'s?
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Non-uniqueness in Krylov-Bogoliubov theorem
Not at all. Take $f(x)=x^2$ on the unit interval. For examples of uniquely ergodic homeomorphisms of the Cantor set see [The prevalence of uniquely ergodic systems](https://mathscinet.ams.org/mathscinet-getitem?mr=252604) by Jewett. A simple explicit example is provided by the boundary action of any hyperbolic automorphism of a homogeneous tree (attach a binary tree to each integer and look at the action of the integer shift on the space of ends of the whole construction). 


