Left-invariant measure

Summary

A left-invariant measure on a group is a measure that gives the same value to a set and to each of its left translates. It is the case of an invariant measure where the maps are the left translations $x \mapsto gx$. The Haar measure of a locally compact group is a left-invariant measure that is also a nonzero Radon measure. Left-invariant measures are the basic tool for integrating over a group without favouring any point.