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)(i) The second moment of $\mu$ is

$$M_{2}(\mu)=\int_{\mathbb{R}^{d}}\lVert x\rVert^{2}\,\mu(dx)\in[0,\infty],$$

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

2. (Finite second moment)(ii) $\mathcal{P}_{2}(\mathbb{R}^{d})$ 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.