Deep learning models have an intrinsic privacy issue as they memorize parts of their training data, creating a privacy leakage. Membership Inference Attacks (MIA) exploit it to obtain confidential information about the data used for training, aiming to steal information. They can be repurposed as a measurement of data integrity by inferring whether it was used to train a machine learning model. While state-of-the-art attacks achieve a significant privacy leakage, their requirements are not feasible enough, hindering their role as practical tools to assess the magnitude of the privacy risk. Moreover, the most appropriate evaluation metric of MIA, the True Positive Rate at low False Positive Rate lacks interpretability. We claim that the incorporation of Few-Shot Learning techniques to the MIA field and a proper qualitative and quantitative privacy evaluation measure should deal with these issues. In this context, our proposal is twofold. We propose a Few-Shot learning based MIA, coined as the FeS-MIA model, which eases the evaluation of the privacy breach of a deep learning model by significantly reducing the number of resources required for the purpose. Furthermore, we propose an interpretable quantitative and qualitative measure of privacy, referred to as Log-MIA measure. Jointly, these proposals provide new tools to assess the privacy leakage and to ease the evaluation of the training data integrity of deep learning models, that is, to analyze the privacy breach of a deep learning model. Experiments carried out with MIA over image classification and language modeling tasks and its comparison to the state-of-the-art show that our proposals excel at reporting the privacy leakage of a deep learning model with little extra information.