  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.
