On convex power series of a conservative Markov operator

A. Brunel proved that a conservative Markov operator, $P$, has a finite invariant measure if and only if every operator $Q = \Sigma _{n = 0}^\infty {\alpha _n}{P^n}$ where ${\alpha _n} \geqq 0$ and $\Sigma {\alpha _n} = 1$ is conservative.

In this note we give a different proof and study related problems.