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



Definition of Haar integral in Bushnell and Henniart
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).


