Left, right, and bi-invariant Borel measures

Let $G$ be a Lie group with identity $e$ (Lie group). A finite Radon measure on $G$ is a Borel measure that is finite on compact sets and inner regular on open sets and outer regular on Borel sets, exactly the Radon convention of Left Haar integral and left Haar measure; it is a probability measure when $\mu (G)=1$. A finite Radon measure $\mu$ on $G$ is

• left invariant when $\mu (gE)=\mu (E)$,
• right invariant when $\mu (Eg)=\mu (E)$,
• inversion invariant when $\mu (E^{-1})=\mu (E)$, and
• bi-invariant when it is both left and right invariant,

for every $g\in G$ and every Borel set $E\subseteq G$.

For a finite Radon measure the set-theoretic conditions above are equivalent to their integral forms: $\mu$ is left invariant if and only if ${\int }_{G}f(a^{-1}x)d\mu (x)={\int }_{G}f(x)d\mu (x)$ for every $a\in G$ and every continuous $f$ on $G$ of compact support, and similarly on the right, with inversion in place of translation for the third condition. Indeed, the translated measure $E\mapsto \mu (aE)$ is again a finite Radon measure, and two finite Radon measures on a locally compact Hausdorff space agree if and only if they give the same integral to every continuous compactly supported function (Radon measure on an LCH space, Uniqueness of the RMK representing measure among Radon measures).

On a compact group the continuous functions are the compactly supported functions, and a finite Radon measure is a probability measure exactly when its integral of the constant function $1$ equals $1$. The normalized Haar measure of a compact Lie group, constructed elsewhere, is the standard example of a bi-invariant probability measure (Normalized Haar measure on a compact Lie group); this definition is the vocabulary in which its invariance is stated.