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][1]


  [1]: https://math.stackexchange.com/questions/2983006/definition-of-haar-integral-in-bushnell-and-henniart