Preliminary Results: The Gaussian Measure and Hermite Polynomials

In this chapter we study the Gaussian measure in \(\mathbb {R}^d\) for d ≥ 1 and several of its properties. Then, we study the problem of the Gaussian measure for balls in \(\mathbb {R}^d,\) which is crucial in Chapter 4 for studying the associated covering lemmas for that measure. For completeness, we consider Hermite polynomials, which are orthogonal polynomials, with respect to the Gaussian measure, and discuss in detail most of their properties. The interested reader will find the properties and identities of all classical orthogonal polynomials listed in the appendix.