In this note, we introduce four measure-theoretic quantities for Borel probability measures to characterize upper and lower Katok measure-theoretic entropies and establish two new variational principles for Bowen topological entropy and packing topological entropy. Besides, we prove that the packing topological entropy of the set of generic points for any invariant ergodic Borel probability measure is equal to the measure-theoretic entropy.

中文翻译：

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关于拓扑和测度理论熵的一些注意事项

在本文中，我们介绍了Borel概率度量的四个度量理论量，以刻画上下Katok度量理论熵的特征，并为Bowen拓扑熵和压缩拓扑熵建立了两个新的变分原理。此外，我们证明了对于任何不变的遍历Borel概率度量，通用点集的堆积拓扑熵等于度量理论熵。