The Homomorphism of Monadic Set-mapping m_T and Standard Part Inverse Mapping st~(-1)

In nonstandard saturated model,the properties of the homomorphism of monadic set-mapping m_T and standard part inverse mapping si~(-1) are discussed.It is proved that a sufficient and necessary condition,under which m_T can be extened to a cr-homomorphism,is that the space X is countably compact.Under some conditions,it is obtained that st~(-1) is aσ-homomorphic mapping on Borel cr-algebra,if and only if the space X is pre-Hausdorff.Finally, Loeb representation of a regular measure is shown.