Let P(.,.) be a probability transition function on a measurable space $(M,\bold M)$ . Let V(.) be a strictly positive eigenfunction of P with eigenvalue ρ>0. Let $$ \widetilde P(x,\d y)\equiv\frac{V(y)P(x,\d y)}{\rho V(x)}. $$ Then $\widetilde P(.,.)$ is also a transition function. Let Px and $\widetilde P_x$ denote respectively the probability distribution of a Markov chain $\{X_j\}^{\infty}_0$ with X0=x and transition functions P and $\widetilde P$ . Conditions for $\widetilde P_x$ to be dominated by Px or to be singular with respect to Px are given in terms of the martingale sequence $W_n\equiv V(X_n)/\rho^n$ and its limit. This is applied to establish an LlogL theorem for supercritical branching processes with an arbitrary type space.