Title: Two Bernoulli shifts with infinite entropy are isomorphic
Abstract: Entropy of fuzzy dynamical systems has been investigated as an isomorphism invariant in Dumitrescu (1995) [13]. In this paper, we will introduce a new invariant called the conditional fuzzy entropy of fuzzy dynamical systems with respect to a finite fuzzy partition and an invariant sub-σ-algebra, which is an extension of fuzzy entropy on fuzzy dynamical systems. This new invariant possesses some basic properties, such as power rule, fuzzy version of Abramov–Rokhlin entropy additive formula and generating sequence.