The problem of causal discovery is especially challenging when the variables of interest cannot be directly measured. In measurement models, the measured variables were generated by latent causal variables that are causally related to each other, and by estimating the measurement model from measured data, one is able to recover the latent variables and their causal relations. In this paper, we provide precise sufficient identifiability conditions for the linear pure measurement model, and show what information of the causal structure can be recovered from observed data without prior knowledge of data distributions. In particular, we first show that, based on second-order statistics, although the pure measurement model is in general not fully identifiable under the assumption of two pure measurement variables, we can identify the set of all candidate measurement models. We then further prove that the pure measurement model can be identified uniquely based on higher-order statistics. Next, to address more general situations, we offer the identifiability conditions of linear measurement models with arbitrary noise distributions. We finally develop a unified method to learn pure measurement models from data. Experimental results on both synthetic and real-world data demonstrate the usefulness of our theory and the effectiveness of our approach.