Gaussian Measure - Properties of Gaussian Measure

Properties of Gaussian Measure

The standard Gaussian measure γn on Rn

• is a Borel measure (in fact, as remarked above, it is defined on the completion of the Borel sigma algebra, which is a finer structure);
• is equivalent to Lebesgue measure:, where stands for absolute continuity of measures;
• is supported on all of Euclidean space: supp(γn) = Rn;
• is a probability measure (γn(Rn) = 1), and so it is locally finite;
• is strictly positive: every non-empty open set has positive measure;
• is inner regular: for all Borel sets A,

so Gaussian measure is a Radon measure;

• is not translation-invariant, but does satisfy the relation

where the derivative on the left-hand side is the Radon–Nikodym derivative, and (T_h)_∗(γn) is the push forward of standard Gaussian measure by the translation map T_h : Rn → Rn, T_h(x) = x + h;

• is the probability measure associated to a normal probability distribution: