Definition of Haar integral in Bushnell and Henniart

Question

In Bushnell and Henniart's The Local Langland's Conjecture for GL(2) they define a right Haar integral on a locally profinite group $G$ as being a non-zero linear functional $$ I: C^{\infty}_{c}(G) \to \mathbb{C}, $$ where the space on the left is the space of all locally constant complex valued functions with compact support on $G$, that satisfies the conditions:

1. $I(\rho_{g}f) = I(f)$ where $\rho_{g}$ is right translation by $g \in G$, $f \in C^{\infty}_{c}(G)$.
2. $I(f) \geq 0$ for $f \geq 0$.

I assume the ordering in the second condition implies that this condition applies to only functions taking real values (otherwise what could it possibly mean), but does this mean that a Haar integral is required to take real valued functions to real numbers, or is it the case that a linear functional on this space must take real values on real functions (possibly conditional on (1))?

Cross-post: MSE

Answer 1

Right, the second condition only really makes sense for real valued functions. In reality, it doesn't matter so much: since a Haar integral is linear it's determined by values on a basis of $C_c^\infty(G)$, and such a basis is given by characteristic functions of open subsets of G. So the condition could just be rephrased as saying it's a right-invariant linear functional with non-negative values on each characteristic function (and the value is the measure of the open subsets under the Haar measure associated to the integral).