Measurable Version of Spectral Decomposition Theorem for a Z2-Action

In this article, we present a measurable version of the spectral decomposition theorem for Z2-action on compact metric space. process, obtain some relationships with shadowing property and k-type weak extending property. Then, introduce definition measure expanding by using properties Borel measure. We also prove one that occurs whenever is invariantly expanding. All supporting results are necessary to theorem, which main result paper. More precisely, if expanding, has property, then it decomposition.