The family of quasi upper Bernoulli shift invariant probability measures is weakly dense in the set of shift invariant probability measures but it is also distinct from weak Gibbs measures. This family of measures include i.i.d. distributions, Markov chains and some hidden Markov chains. They also naturally emerge as models of repeated measurements of quantum systems. This last application motivated the work I will present. With N. Cuneo, V. Jaksic, Y. Pautrat and C.-A. Pillet, we studied hypothesis testing for these measures and we were particularly interested in its application to thermodynamics for repeated quantum measurements. I will present part of our work. After an exposition of our physical motivations I will explain our proof of identifiability of the measures and equality of Stein's exponent. I will then explain how using sub additive thermodynamic formalism we can obtain differentiability of Rényi's relative entropy and equality of Hoeffding's exponents and equality of Chernoff's ones. I will finish the presentation with some examples highlighting the richness of this family of measures (connexion to number theory, phase transitions for 1D spin chains ...). Ref: CMP 2018 arXiv:1607.00162