Godambe (1955) give a general finite population sampling model and proved that a best linear unbiased estimator (BLUE) of population total does not exist when there is no measurement error. In this research, Godambe's linear estimator is expanded to include two types of measurement errors and their mixture. We check Godambe's non-existence theorem and explore the method to develop the best linear unbiased estimator of the latent population total by using individual unbiased constraints and average unbiased constraints, respectively. We start from Godambe's general framework and then reduce to two probability models which are less general than Godambe's. The model is developed under unequal probability sampling without replacement. As a special case, the model under simple random sampling without replacement can be derived directly based on the results. The traditional definition of inclusion probability is extended and applied to the unequal probability sampling. These results connect the traditional sampling method and the design-based method using random permutation models based on the work of Royall (1976) as proposed by Stanek, Singer and Lencina (2004). We also examine the relationship among Godambe's general finite sampling model, the expanded model finite population model and the finite population mixed model. Also, we are able to give another set of solutions by giving a distribution to the sample latent values. The research can serve as the basis for extensions to multi-stage sampling or other complex sampling designs.