Definitions
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 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.
Read more about this topic: Gaussian Measure
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