The Entropy of a Nonnegative Function with Respect to a Probability Measure

In the setting of Measure Spaces and the Lebesgue Integral: Standing Notation, let $(\Omega,\mathcal{F},\mu)$ be a measure space with $\mu(\Omega)=1$, and let $\phi:[0,\infty)\to\mathbb{R}$ be the function of The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm, with $\phi(0)=0$ and $\phi(s)=s\log s$ for positive $s$.

(Entropy of a function)¶ Let $G:\Omega\to\mathbb{R}$ be measurable with $0\le G(\omega)$ for every $\omega\in\Omega$, such that $G$ and $\phi\circ G$ are integrable with respect to $\mu$; $\phi\circ G$ is measurable by The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §continuous, and $\int_{\Omega}G\,d\mu\ge0$ by the monotonicity of Linearity and Monotonicity of the Lebesgue Integral §nonnegative. The entropy of $G$ with respect to $\mu$ is the real number

$$\operatorname{Ent}_{\mu}(G)=\int_{\Omega}\phi\circ G\,d\mu-\phi\Bigl(\int_{\Omega}G\,d\mu\Bigr).$$