Left Haar integral and left Haar measure

Let $G$ be a group with a Hausdorff locally compact topology for which multiplication $G\times G\to G$ and inversion $G\to G$ are continuous. The identity $e$ belongs to $G$, so $G$ is nonempty. Use Group and abelian group, Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space and Compact support, $C_c(X)$, and $C_0(X)$. Set $L_{a}f(x)=f(a^{-1}x)$ and $R_{a}f(x)=f(xa)$.

A left Haar integral is a nonzero positive real-linear functional $I:C_c(G;\mathbb{R})\to \mathbb{R}$ such that $I(L_{a}f)=I(f)$ for every $a,f$; positivity has the meaning in Positive linear functionals on $C_c(X)$. A left Haar measure is a nonzero Borel measure $\mu$ with $\mu (aE)=\mu (E)$ for all Borel $E$ and all $a$, finite on compact sets, outer regular on Borel sets and inner regular on open sets, exactly as in Radon measure on an LCH space. Right Haar replaces left translations by right translations. These are independent definitions; existence and their correspondence are established later. Complexification means $I(u+iv)=I(u)+iI(v)$ for real $u,v$. No countability assumption on $G$ is imposed.

Pedersen, Haar integral, p.2 definitions and lemma; p.3 Theorem 1; pp.4–5 second proof and Remark 2. Local argument and conventions as displayed above.