This paper develops a non-periodic inspection maintenance model for a gamma deteriorating single-unit system based on regenerative and semi- regenerative process theory. Minimizing the expected average cost per unit time is chosen as the optimization goal. The model uses an inspection scheduling function to arrange the inspection scheme based on the system state. Considering measurement errors make the model more realistic. The preventive maintenance threshold and the inspection scheme are obtained after the optimization. A numerical example is given in the last and proves the effectiveness of the model by the sensitivity analysis of degradation rate and cost parameters. By the comparison with the optimal results without considering measurement errors, it proves that measurement error is not a trivial thing for the maintenance optimization. 