Existence of a left Haar integral

Take a nonzero nonnegative cutoff at the identity and a common point $T$ from [F2]. That lemma gives $T(u+v)=T(u)+T(v)$ for all nonnegative $u,v$; it is positively homogeneous, invariant, and $T(f_0)=1$.

Every real $f\in C_c(G)$ is $u-v$ with $u=f^{+}$ and $v=f^{-}$. Define $I(f)=T(u)-T(v)$. If also $f=u^{\mathrm{\prime }}-v^{\mathrm{\prime }}$ with nonnegative terms, then $u+v^{\mathrm{\prime }}=u^{\mathrm{\prime }}+v$, so additivity gives $T(u)-T(v)=T(u^{\mathrm{\prime }})-T(v^{\mathrm{\prime }})$. The value is independent of the decomposition.

Adding decompositions and multiplying them by a nonnegative scalar proves additivity and positive homogeneity of $I$; exchanging their terms gives $I(-f)=-I(f)$, hence real linearity. For $f\ge 0$ use $f-0$ to obtain $I(f)=T(f)\ge 0$. Translating both nonnegative terms gives $I(L_{a}f)=I(f)$; $I(f_0)=1$ makes $I$ nonzero. Finally $I_{\mathbb{C}}(u+iv)=I(u)+iI(v)$ is well-defined by unique real and imaginary parts, and $I_{\mathbb{C}}(i(u+iv))=-I(v)+iI(u)=i{I}_{\mathbb{C}}(u+iv)$ proves complex linearity together with real linearity.