Abstract: In the paper, by using the bounded $\Phi$-variation function,the stability of the bounded $\Phi$-variation solution to measure functional differential equations with infinite delay is discussed. With respect to this kind of equations, the $\Phi$-variational stability, the $\Phi$-variational attraction and asymptotically $\Phi$-variational stability are defined. The Lyapunov type theorems for the $\Phi$-variational stability and asymptotically $\Phi$-variational stability of the bounded $\Phi$-variation solutions are established. These results are essential generalization of the current variational stability theorem for measure functional differential equations with infinite delay.

Key words: measure functional differential equations with infinite delay, bounded $\Phi$-variation solution, $\Phi$-variational stability, asymptotically $\Phi$-variational stability, Lyapunov function