  For a $*$-automorphism group $G$ on a $C^*$- or von Neumann algebra, we study
the $G$-quasi invariant states and their properties. The $G$-quasi invariance
or $G$-strongly quasi invariance are weaker than the $G$-invariance and have
wide applications. We develop several properties for $G$-strongly quasi
invariant states. Many of them are the extensions of the already developed
theories for $G$-invariant states. Among others, we consider the relationship
between the group $G$ and modular automorphism group, invariant subalgebras,
ergodicity, modular theory, and abelian subalgebras. We provide with some
examples to support the results.
