We introduce average shadowable measures and almost average shadowable measures for continuous maps of compact metric spaces and weakly topologically stable points and average persistent property for homeomorphisms of compact metric spaces. We prove that the set of all average shadowable measures is dense in the space of all Borel probability measures if and only if the set of all average shadowable points is dense in the phase space and every almost average shadowable measure can be weak^⁎ approximated by measures having support equal to the closure of the set of all average shadowable points. Moreover, we prove that every minimally expansive point which is either persistent or α-persistent is weakly topologically stable and every mean equicontinuous pointwise weakly topologically stable homeomorphism is average persistent.

对于紧测度空间的连续映射和弱测度稳定点以及紧测度空间的同胚性的平均持久性，我们介绍了平均可影子测度和几乎平均可影子测度。我们证明了集合的所有平均shadowable措施密集在所有的Borel概率测度的空间，当且仅当集合中的所有平均shadowable点是密集的相空间中，每几乎平均shadowable措施可能很弱^⁎采取措施近似具有等于​​关闭所有平均可阴影点集的支持。而且，我们证明了每个最小的扩张点，无论是持久的还是α-持久是弱拓扑稳定的，并且每个均等点连续的点向弱拓扑稳定同胚是平均持久的。