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Let $X$ be a compact metric space and $T$ a continuous map from $X$ to $X$. The system $(X,T)$ is called topologically weakly mixing if the product system $(X\times X,T\times T)$ is topologically transitive. My question is that: If $(X,T)$ is minimal and topologically weakly mixing, whether there exists an ergodic measure $\mu$ on $(X,T)$ such that $(X,T)$ is weakly mixing as a measure-preserving system, which means $(X\times X,T\times T,\mu\times \mu)$ is ergodic?

I guess there might be a counter-example but I do not have any such example in mind, nor can prove it. If this is not true, is it possible to add some proper condition to make this true but not trivially true?

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    $\begingroup$ I am sure the answer is no, but I can’t think of an immediate counterexample. Basically these topological notions are not able to detect whether there is a measure-theoretic rotation factor (which is what lack of weak mixing is). $\endgroup$ Feb 29 '20 at 21:10
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Now I figure out that the answer of my question is false. In the paper ``Topological mixing and uniquely ergodic systems", Lehrer proved any measure-preserving system has a topologcally mixing strictly ergodic topological model, which implies that there exists a topologcally mixing strictly ergodic system $(X,T)$ such that with the unique invariant measure $\mu$, is measure-theoretically isomorphic to an irrational rotation on the circle, which is not measure weakly mixing. So my question is even false when topologically weakly mixing is replaced by topologcally mixing .

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