Left-invariant measure

Remarks

The set $gA$ is again a Borel set, since left translation by $g$ is a homeomorphism of $G$. Equivalently, the pushforward of $\mu$ under each left translation $x \mapsto gx$ equals $\mu$. In integral form, $\int_G f(gx) \, d\mu(x) = \int_G f(x) \, d\mu(x)$ for every $g \in G$ and every nonnegative Borel function $f$.

Example: Lebesgue measure $\lambda$ on $(\R, +)$. Here $gA = g + A$, and $\lambda(g + A) = \lambda(A)$ for every Borel set $A$, since length does not change under shifts.

Non-example: the Dirac measure $\delta_0$ on $(\R, +)$. One has $\delta_0(\set{0}) = 1$ but $\delta_0(1 + \set{0}) = \delta_0(\set{1}) = 0$.

The zero measure and the counting measure are left-invariant on every group, but they are not Haar measures on $\R$. The first is zero, and the second is infinite on infinite compact sets. A Haar measure is a left-invariant measure that is a nonzero Radon measure.