Finite entropy chain rule and monotonicity under conditioning

On every positive-probability block, factor its probability into successive conditional probabilities by repeated use of $P(A\cap B)=P(A)P(B\mid A)$. Taking $-{\log }_2$ and averaging over the finite support yields $H(X_0^{n-1})={\sum }_{i=0}^{n-1}H(X_i\mid X_0^{i-1})\mathrm{.}$ The $i=0$ term is $H(X_0)$.

For finite random variables $U,V$, the same factorization gives $H_2(U,V)=H_2(U)+H_2(V\mid U)$. Terms with $P(U=u)=0$ contribute zero, so no conditional law on a null event affects the identity.

For finite $U,V,W$, fix any $v$ with $P(V=v)>0$. The conditional mutual information at that value is ${\sum }_{w}P(w\mid v)D_2(P_{U\mid v,w}\mathrm{\parallel }{P}_{U\mid v})\ge 0$ by [F2]; if a conditional reference probability is zero, every positive-weight conditional law also assigns that outcome zero, so all displayed divergences are defined on their supports. Averaging over $v$ gives $H_2(U\mid V)-H_2(U\mid V,W)\ge 0$.