An invariant measure is called a Bernoulli measure if the corresponding dynamics is  isomorphic to a Bernoulli shift. We prove that for  C^{1+α} diffeomorphisms any weak-mixing hyperbolic measure could be approximated by Bernoulli measures. This also holds true for  C^1 diffeomorphisms preserving a weak-mixing  hyperbolic measure and  the Oseledets decomposition with respect to the measure is dominated.

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