I'm reading Viana&Oliveira's Book, Foundations in Ergodic Theory, and in viewing of Lemma in the image I cannot understand why in the proof, the set $\mathcal{S}$ is non-empty. Someone can explain me why?. I attach the image
Edit : I thought about this, and my approach is : In this case, if $\mathcal{S}$ is empty, you can construct a partition $\mathcal{Q}_1$ such that every element is of the form $Q_1\cup Q_2$, with $Q_i \in \mathcal{Q}$. Then you construct an $\mathcal{S}_1$. By induction and in finite steps, you can construct the desire sets $\mathcal{S}_j \subset \mathcal{Q}_j$, with $\mathcal{S}_j$ non empty, this is due by using the fact that we are working with partitions, i.e. in some time we can cover $P$ with elements of $\mathcal{Q}$ (up to measure 0). This with properties of entropy of a partition under the relation $\prec$ we can see that this construction does not affect the argument.