On the set of shadowable measures

We first prove that the set of shadowable measures of a homeomorphism of a compact metric space X is an Fσδ set of the space M(X) of Borel probability measures of X equipped with the weak*-topology. Next that the set of shadowable measures is dense in M(X) if and only if the set of shadowable points is dense in X. Therefore, if X has no isolated points and every non-atomic Borel probability measure has the shadowing property, then the shadowable points are dense in X (this is false when the space has isolated points). Afterwards, we consider the almost shadowable measures (measures for which the shadowable point set has full measure) and prove that all of them are weak* approximated by shadowable ones. In addition the set of almost shadowable measures is a Gδ set of M(X). Furthermore, the closure of the shadowable points is the union of the supports of the almost shadowable measures. Finally, we prove that every almost shadowable measure can be weak* approximated by ones with support equals to the closure of the shadowable points.