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For a measurable map $T$ and a sequence of $T$-invariant probability measures$\mu_n$ that converges in some sense to a $T$-invariant probability measure$\mu$, an estimate from below for the Kolmogorov--Sinai entropy of $T$ withrespect to $\mu$ is suggested in terms of the entropies of $T$ with respect to$\mu_1$, $\mu_2$, \dots.

Weak approximation of an invariant measure and a low boundary of the entropy