The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $d\in\mathbb{N}$ satisfy $1\le d$, and let $\mu\in\mathcal{P}(\mathbb{R}^{d})$.

1. (Second moment)¶ The second moment of $\mu$ is

the integral, in the sense of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures, of the nonnegative Borel function $x\mapsto\lVert x\rVert^{2}$ of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs.

2. (Finite second moment)¶ $\mathcal{P}_{2}(\mathbb{R}^{d})$ denotes the set of all $\mu\in\mathcal{P}(\mathbb{R}^{d})$ with $M_{2}(\mu)<\infty$, the probability measures on $\mathbb{R}^{d}$ with finite second moment.