How do we deal the discrete case? For example, in directional light what is the p(w) for each direction? If p(w) is 1 for the specific solid angle (and 0 elsewhere), then how could the integration of p(w) to be 1? Moreover, what is p(w) if a certain light has two fixed direction? Is p(w)=0.5 for these two specific directions?

How do we deal the discrete case? For example, in directional light what is the p(w) for each direction? If p(w) is 1 for the specific solid angle (and 0 elsewhere), then how could the integration of p(w) to be 1? Moreover, what is p(w) if a certain light has two fixed direction? Is p(w)=0.5 for these two specific directions?