Normalized Haar measure on a compact Lie group

A finite-dimensional Lie group is a Hausdorff topological group, and compactness is a topological property, so a compact Lie group is a compact Hausdorff group; by [L1] it therefore carries a unique left Haar probability measure $\mu$, which is right invariant and inversion invariant.

Since $G$ is compact, every left Haar measure on $G$ has finite positive total mass and can be normalized by dividing by that mass; compactness alone does not make an arbitrary Haar measure a probability measure. For the normalized measure, outer regularity also gives inner regularity on every Borel $E$: for $ϵ>0$ choose open $U\supseteq G\setminus E$ with $\mu (U)<\mu (G\setminus E)+ϵ$; then $C=G\setminus U$ is compact, $C\subseteq E$, and $\mu (C)>\mu (E)-ϵ$. Conversely, a regular Borel probability measure that is left invariant is a left Haar measure of total mass one in the sense of [L2]. Hence the uniqueness assertion of [L1] is exactly uniqueness among regular Borel probability measures that are left invariant.

By [L3] the measure $\mu$ of step 1.1 is bi-invariant satisfying all three invariance conditions, so a measure with the stated properties exists.

For uniqueness let $\nu$ be any regular Borel probability measure invariant under left and right translations and inversion. Then $\nu$ is a left Haar probability measure, so $\nu =\mu$ by the uniqueness in [L1], applied through step 1.2.

Existence and uniqueness are established, and the Axiom of Choice entered exactly through the uniqueness-and-existence statement [L1].