Approximating a non-negative probability measure by another

Question

$l_{1,+}(G) = \{f \in l^{1}(G): ||f||_{1}= 1,f\geq 0\}$, where $G$ is a finitely generated group. I have to prove that every finitely supported function in this set can be approximated by another function in the same set but its codomain will be $\{0,\frac{1}{N}, \frac{2}{N},....1\}$ for some natural number $N$ i.e., for any $\epsilon > 0$, there exists $g \in l_{1,+}(G) $ with aforesaid codomain such that $||f - g|| < \epsilon $. I thought of considering small neighbourhoods of elements in the image of the given function but couldn't find something that satisfies all conditions. Another thing that comes to my mind is something similar to a convolution but I don't know how to define such in the case of a group.