Finite f-divergence with support conventions

Let $P=(p_x)$ and $Q=(q_x)$ be probability vectors on a finite nonempty alphabet $\mathcal{X}$ (Finite probability vectors). A convex generator is a finite convex function $f:(0,\mathrm{\infty })\to \mathbb{R}$ with $f(1)=0$; convexity uses the two-point inequality of Convex and strictly convex functions on Euclidean convex sets. Use its extended boundary values $f(0):={\lim }_{t\downarrow 0}f(t),f_{\mathrm{\infty }}:={\lim }_{t\to \mathrm{\infty }}\frac{f(t)}{t}\mathrm{.}$ These limits exist in $\mathbb{R}\cup \{+\mathrm{\infty }\}$ by Boundary limits of a normalized finite convex generator exist. Define the finite f-divergence from $P$ to $Q$ by $D_f(P\mathrm{\parallel }Q):={\sum }_{x:q_x>0}{q}_{x}f(p_x\mathrm{/}{q}_x)+f_{\mathrm{\infty }}{\sum }_{x:q_x=0}{p}_x\mathrm{.}$ In the first sum, if $p_x=0$, the term means $q_{x}f(0)$. A coordinate with $p_x=q_x=0$ contributes zero and is omitted. The second term is defined as zero when its coefficient ${\sum }_{q_x=0}{p}_x$ is zero, even if $f_{\mathrm{\infty }}=+\mathrm{\infty }$; if that coefficient is positive and $f_{\mathrm{\infty }}=+\mathrm{\infty }$, the divergence is $+\mathrm{\infty }$.

The boundary term is essential when $P\ll ̸Q$. It is the recession contribution of the generator and is not the value $f(0)$.