Proof degenerating discrete probability.

Question

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?

Answer 1

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 $\mu$