Invariant measures for $\mathscr{B}$-free systems revisited

For $\mathscr{B} \subseteq \mathbb{N}$, the $\mathscr{B}$-free subshift $X_{\eta}$ is the orbit closure of the characteristic function of the set of $\mathscr{B}$-free integers. We show that many results about invariant measures and entropy, previously only known for the hereditary closure of $X_{\eta}$, have their analogues for $X_{\eta}$ as well. In particular, we settle in the affirmative a conjecture of Keller about a description of such measures ([Keller, G. Generalized heredity in $\mathcal B$-free systems. Stoch. Dyn. 21, 3 (2021), Paper No. 2140008]). A central assumption in our work is that $\eta^{*}$ (the Toeplitz sequence that generates the unique minimal component of $X_{\eta}$) is regular. From this we obtain natural periodic approximations that we frequently use in our proofs to bound the elements in $X_{\eta}$ from above and below.