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

In the setting of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation, the function $x\mapsto|x|^{2}$ on $X$ is nonnegative and continuous, since $x\mapsto|x|$ is Lipschitz with constant $1$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity and products of continuous real functions are continuous by Continuity of Sums and Products of Real-Valued Functions on a Metric Space; hence it is Borel by claims 2 and 3 of Borel Measurability and Bounded Integration on a Metric Space, and its integral against every $\mu\in\mathcal{P}(X)$ is defined in $[0,\infty]$ by Measure Spaces and the Lebesgue Integral: Standing Notation §integral.

1. (Second moment)¶ The second moment of $\mu\in\mathcal{P}(X)$ is

$$M_{2}(\mu)=\int_{X}|x|^{2}\,\mu(dx)\in[0,\infty].$$

2. (Finite second moment)¶ $\mathcal{P}_{2}(X)$ is the set of all $\mu\in\mathcal{P}(X)$ with $M_{2}(\mu)<\infty$, the Borel probability measures on $X$ with finite second moment.