Society is increasingly relying on predictive models in fields like criminal justice, credit risk management, or hiring. To prevent such automated systems from discriminating against people belonging to certain groups, fairness measures have become a crucial component in socially relevant applications of machine learning. However, existing fairness measures have been designed to assess the bias between predictions for protected groups without considering the imbalance in the classes of the target variable. Current research on the potential effect of class imbalance on fairness focuses on practical applications rather than dataset-independent measure properties. In this paper, we study the general properties of fairness measures for changing class and protected group proportions. For this purpose, we analyze the probability mass functions of six of the most popular group fairness measures. We also measure how the probability of achieving perfect fairness changes for varying class imbalance ratios. Moreover, we relate the dataset-independent properties of fairness measures described in this paper to classifier fairness in real-life tasks. Our results show that measures such as Equal Opportunity and Positive Predictive Parity are more sensitive to changes in class imbalance than Accuracy Equality. These findings can help guide researchers and practitioners in choosing the most appropriate fairness measures for their classification problems.