Invariant measures is a central concept in the study of dynamical systems and group actions, referring to measures defined on a space that are preserved under the transformations or group elements acting on that space. This means the measure of any measurable set is equal to the measure of its image under the action. The concept investigates systems where a distribution or 'size' on the state space remains statistically stable over time or under group operations. Key characteristics include the property that the measure of a transformed set equals the measure of the original set for all relevant transformations and measurable sets, built upon the standard properties of measures such as countable additivity. Their significance lies in providing a fundamental framework for analyzing the long-term average behavior of systems and serving as the foundation for ergodic theory, particularly in defining ergodicity and classifying transformations based on their measure-preserving properties, although existence is not guaranteed in all settings and related concepts like quasi-invariant measures may be used.