In this paper, the Gerber-Shiu function for a discrete risk models is considered for the surplus of an insurance company mainly under some light-tail assumptions. By a change of measure we remove the discounting, which simplifies the expression, and this leads to (defective) renewal equation that had been found. By the new Gerber-Shiu function, the Lundberg inequalities can easily be obtained. If we use the change of measure such that ruin becomes certain, the renewal equations simplify to ordinary renewal equations. This helps to discuss the asymptotic as the initial capital goes to infinity. For exponential distribution claim sizes, explicit formula for the ruin probability can be derived.