If $C$ is a compact (semi-)topological (semi-)group, are there nonzero positive functions having zero Haar integral? In other words: is the Hermitian product associated to the Haar integral degenerate?

The motivation of my question: if $G$ is a locally-compact (semi-)topological (semi-)group, $C$ its almost periodic (AP) compactification and $\lambda$ a Haar measure on $G$, then for any AP function $f$ on $G$ and any Foelner sequence $H_n$ in $G$ one has:

$\int_{C} f \mathbb{dc} = \mathbb{lim}_{n\rightarrow\infty} \frac{\int_{H_n} f \mathbb{dg}}{\lambda (H_n)}$

But then, if $f$ is as above AND integrable (like $\frac{1}{1+x^2}$ on $R$), the previous result implies that when it is lifted to the compatification its integral becomes 0 (despite the fact that its lift is positive, since a group is dense in its compactification).

For details about the theorem see Hewitt & Ross, "Abstract Harmonic Analysis", p.252-253.