  We study the stability of $\mathcal{M}_0$, an invariant subset of a Markov
process $(X_t)_{t\geq 0}$ on a metric space $\mathcal{M}$. By building the
theory of average Lyapunov functions, we formulate general criteria based on
the signs of Lyapunov exponents that guarantee extinction ($X_t \to
\mathcal{M}_0$ as $t \to \infty$). Additionally, we provide applications to a
stochastic SIS epidemic model on a network with regime-switching, a stochastic
differential equation version of the Lorenz system, a general class of
discrete-time ecological models, and stochastic Kolmogorov systems. In many
examples we improve existing results by removing unnecessary assumptions or
providing sharper criteria for the extinction.
