Definition (Dirac distribution)

Fix $i\in S$. Denote $\delta_i$ as the pmf so that $$\delta_i(j)=\left\{ \begin{array}{ll} 1 & \mbox{if }j=i\\ 0 & \mbox{otherwise} \end{array}\right.$$$\delta_i$ is known as the Dirac distribution at state $i$. Thus is RV $X$ has a distribution $\delta_i$, it means $$P(X=i)=1$$In other words, w.p.1. $X$ takes value $i$.