Local martingales in discrete time

For any discrete-time P-local martingale S there exists a probability measure Q similar to P such that S is a Q-martingale. A new proof for this result is provided. The core idea relies on an appropriate modification of an argument by Chris Rogers, used to prove a version of the fundamental theorem of asset pricing in discrete time. This proof also yields that, for any epsilon > 0, the measure Q can be chosen so that dQ/dP <= 1 + epsilon.