Is there a class of metric measure spaces for which a probability measure is preserved by an isometry? It is clear that for a generic metric measure space, isometries do not preserve measure. However, it intuitively makes sense to think that for instance, the Euclidean group preserves probability measure.
Is there a proof to show that either this is or is not the case? Moreover, if this is true, is there a larger class of metric measure spaces for which this is true?
I am new to this field so pardon the lack of rigor in asking the question.