The Entropy of a Probability Measure on Euclidean Space

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $d$ be a natural number with $1\le d$, and let $\lambda_{d}$ be Lebesgue measure on $\mathcal{B}(\mathbb{R}^{d})$. Densities with respect to $\lambda_{d}$ are those of The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities, $\phi$ is the function $s\mapsto s\log s$ of The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm, and $\mathcal{P}_{2}(\mathbb{R}^{d})$ is the set of probability measures with finite second moment. Let $\mu\in\mathcal{P}(\mathbb{R}^{d})$. For a density $\rho$ of $\mu$ with respect to $\lambda_{d}$ the function $\phi\circ\rho$ is Borel by The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §continuous.

(Entropy)¶ The measure $\mu$ has finite entropy if it has a density $\rho$ with respect to $\lambda_{d}$ for which $\phi\circ\rho$ is integrable with respect to $\lambda_{d}$. The entropy of such a $\mu$ is the real number

It does not depend on the choice of $\rho$: two densities of $\mu$ with respect to $\lambda_{d}$ agree outside a set of $\lambda_{d}$-measure $0$ by The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities §uniqueness, and the integrals agree by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison. The set of $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ with finite entropy is denoted $\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$.