ThmDex – An index of mathematical definitions, results, and conjectures.
Proof P3321 on R4844: Mutual information of a simple random variable with respect to itself
P3321
Using results
(i) R4843: Conditional entropy formula for simple mutual information
(ii) R4846: Conditional simple entropy with respect to itself

we have \begin{equation} I_a(X ; X) = H_a(X) - H_a(X \mid X) = H_a(X) - 0 = H_a(X) \end{equation} $\square$