A common approach to explaining NLP models is to use importance measures that express which tokens are important for a prediction. Unfortunately, such explanations are often wrong despite being persuasive. Therefore, it is essential to measure their faithfulness. One such metric is if tokens are truly important, then masking them should result in worse model performance. However, token masking introduces out-of-distribution issues, and existing solutions that address this are computationally expensive and employ proxy models. Furthermore, other metrics are very limited in scope. This work proposes an inherently faithfulness measurable model that addresses these challenges. This is achieved using a novel fine-tuning method that incorporates masking, such that masking tokens become in-distribution by design. This differs from existing approaches, which are completely model-agnostic but are inapplicable in practice. We demonstrate the generality of our approach by applying it to 16 different datasets and validate it using statistical in-distribution tests. The faithfulness is then measured with 9 different importance measures. Because masking is in-distribution, importance measures that themselves use masking become consistently more faithful. Additionally, because the model makes faithfulness cheap to measure, we can optimize explanations towards maximal faithfulness; thus, our model becomes indirectly inherently explainable.