Haar measure is a unique measure defined on locally compact topological groups that is left-invariant, meaning it assigns the same measure to sets that can be transformed by group translations. This concept is fundamental in ergodic theory and harmonic analysis, as it allows for the integration of functions over groups in a way that respects the group structure. Haar measure ensures that when studying dynamical systems, one can apply the concepts of probability and integration consistently across transformations of the system.
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