No-arbitrage and equivalent martingale measures: an elementary proof of the Harrison–Pliska theorem

We give a new proof of a key result to the theorem that in the discrete-time stochastic model of a frictionless security market the absence of arbitrage possibilities is equivalent to the existence of a probability measure $Q$ which is absolute continuous with respect to the basic probability measure $P$ with the strictly positive and bounded density and such that all security prices are martingales with respect to $Q$. The proof is elementary in a sense that it does not involve a measurable selection theorem.