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Given a substochastic kernel $P$ from a measurable space $(E,\beta )$ into itself one considers for a pair $(\mu ,\nu )$ of finite measures on $\beta$ the following sequences: $${\mu }_0=(\mu -\nu {)}^{+},{\nu }_0=(\mu -\nu {)}^{-};$$ $${\mu }_{n+1}=({\mu }_n^p-{\nu }_n{)}^{+},{\nu }_{n+1}=({\mu }_n^p-{\nu }_n{)}^{-},n=0,1,2,...$$ This paper deals with conditions for $\downarrow {lim}_n{\nu }_n=0$, or ${lim}_n\parallel {\mu }_n\parallel =0$ to hold. As an application a characterization of those measures $\nu$ is given which may occur in a $P$-Markov chain $(X_n{)}_{n\in \mathbf{N}}$ with state space $E$, having $\mu$ as its initial law, as distribution of $X_{\tau }$ where $\tau$ is a suitable stopping time.