Proof degenerating discrete probability. 
I'm finding the yellow square confusing in this proof, why $X=E[X]$? I mean indirectly it makes sense with the $c=\mu$, but how do they directly equate? 

If $\exists c \in \mathbb{R}$ s.t. $P(X = c) = 1$, then $E[X] = E[c] = c$.
I think the order is quite confusing. I think better is
$$X = c = E[X] := \mu,$$
which is to say: $X$ is a constant $c$ and hence is equal to its expected value which we denote $\mu$
