Gaussian measure

In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space Rn, closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are named after the German mathematician Carl Friedrich Gauss. One reason why Gaussian measures are so ubiquitous in probability theory is the central limit theorem. Loosely speaking, it states that if a random variable X is obtained by summing a large number N of independent random variables of order 1, then X is of order N {displaystyle {sqrt {N}}} and its law is approximately Gaussian. Let n ∈ N and let B0(Rn) denote the completion of the Borel σ-algebra on Rn. Let λn : B0(Rn) → denote the usual n-dimensional Lebesgue measure. Then the standard Gaussian measure γn : B0(Rn) → is defined by for any measurable set A ∈ B0(Rn). In terms of the Radon–Nikodym derivative, More generally, the Gaussian measure with mean μ ∈ Rn and variance σ2 > 0 is given by Gaussian measures with mean μ = 0 are known as centred Gaussian measures. The Dirac measure δμ is the weak limit of γ μ , σ 2 n {displaystyle gamma _{mu ,sigma ^{2}}^{n}} as σ → 0, and is considered to be a degenerate Gaussian measure; in contrast, Gaussian measures with finite, non-zero variance are called non-degenerate Gaussian measures. The standard Gaussian measure γn on Rn so Gaussian measure is a Radon measure;