The concept of shadowable measures in dynamical systems is well studied. This paper aims to generalize Shin’s (J. Math. Anal. Appl. 469(2), 872–881 (2019)) main results to free semigroup actions. Specifically, we introduce the notion of shadowable measures for a free semigroup action. Let \( G \) be the free semigroup generated by finite continuous self-maps acting on a compact metric space \( X \) . We prove the following results: (1) The set of shadowable measures of \( G \) is an \( F_{\sigma \delta } \) set in the space \( \mathscr {M}(X) \) of Borel probability measures on \( X \) , equipped with the weak*-topology. (2) The set of shadowable measures of \( G \) is dense in \( \mathscr {M}(X) \) if and only if the set of shadowable points of \( G \) is dense in \( X \) . (3) Almost shadowable measures of \( G \) can be weak*-approximated by shadowable measures. (4) The set of almost shadowable measures of \( G \) is a \( G_{\delta } \) set in \( \mathscr {M}(X) \) . (5) The closure of the set of shadowable points of \( G \) is the union of the supports of the almost shadowable measures of \( G \) . (6) Every almost shadowable measure of \( G \) can be weak*-approximated by measures whose support equals the closure of the shadowable points of \( G \) .
