Haar measure

Remarks

Example: Lebesgue measure $\lambda$ on $(\R^n, +)$. It is finite on compact sets, regular and nonzero, and $\lambda(x + A) = \lambda(A)$ for every $x \in \R^n$ and Borel set $A$.

Non-example: the Dirac measure $\delta_0$ on $(\R, +)$ is a nonzero Radon measure but fails clause 3, since $\delta_0(\set{0}) = 1$ and $\delta_0(1 + \set{0}) = \delta_0(\set{1}) = 0$.

In integral form, left invariance says $\int_G f(gx) \, d\mu(x) = \int_G f(x) \, d\mu(x)$ for every $g \in G$ and every compactly supported continuous $f$. If $\mu$ is a left Haar measure, then $A \mapsto \mu(A^{-1})$ is a right Haar measure. A left Haar measure gives positive measure to every nonempty open set.