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For nonnegative densities of possibly different total masses, this divergence includes the mass correction $-x+y$ as well as the logarithmic term. It is the Bregman distance of the maximum-entropy regularization functional at a positive reference $y$. Convexity of $s\log s-s$ proves nonnegativity. For equal masses the correction integrates to zero, recovering the Kullback-Leibler divergence; for probability density functions both masses are one. Use the usual extended-value conventions at zero reference density.

For the maximum-entropy regularization functional, differentiation of the scalar integrand gives $(s\log s-s{)}^{\prime }=\log s$. Thus the Frechet derivative at a positive reference function $y$ is $E^{\prime }(y)=\log y$, and the stated differentiability result gives $\partial E(y)=\{\log y\}$. Consequently,This is the generalized Kullback–Leibler divergence. When both functions integrate to the same mass, the last two terms cancel after integration; for probability density functions it becomes the usual Kullback-Leibler divergence. For two positive functions, adding the reverse distance givesA sufficient setting for the differentiability argument is $L^{\infty }(\Omega )$ with $y$ bounded away from zero. For nonnegative $x$, use $0\log (0/y)=0$ wherever appropriate. Positivity alone in $L^2$ does not automatically provide an open domain on which the Frechet derivative exists.