Let n ∈ N and let B₀(Rⁿ) denote the completion of the Borel σ-algebra on Rⁿ. Let λⁿ : B₀(Rⁿ) → [0, +∞] denote the usual n-dimensional Lebesgue measure. Then the standard Gaussian measure γⁿ : B₀(Rⁿ) → [0, 1] is defined by

${\displaystyle \gamma ^{n}(A)={\frac {1}{{\sqrt {2\pi }}^{n}}}\int _{A}\exp \left(-{\frac {1}{2}}\|x\|_{\mathbb {R} ^{n}}^{2}\right)\,\mathrm {d} \lambda ^{n}(x)}$ 

for any measurable set A ∈ B₀(Rⁿ). In terms of the Radon–Nikodym derivative,

${\displaystyle {\frac {\mathrm {d} \gamma ^{n}}{\mathrm {d} \lambda ^{n}}}(x)={\frac {1}{{\sqrt {2\pi }}^{n}}}\exp \left(-{\frac {1}{2}}\|x\|_{\mathbb {R} ^{n}}^{2}\right).}$ 

More generally, the Gaussian measure with mean μ ∈ Rⁿ and variance σ² > 0 is given by

${\displaystyle \gamma _{\mu ,\sigma ^{2}}^{n}(A):={\frac {1}{{\sqrt {2\pi \sigma ^{2}}}^{n}}}\int _{A}\exp \left(-{\frac {1}{2\sigma ^{2}}}\|x-\mu \|_{\mathbb {R} ^{n}}^{2}\right)\,\mathrm {d} \lambda ^{n}(x).}$ 

Gaussian measures with mean μ = 0 are known as centred Gaussian measures.

The Dirac measure δ_μ is the weak limit of ${\displaystyle \gamma _{\mu ,\sigma ^{2}}^{n}}$  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

• 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: ${\displaystyle \lambda ^{n}\ll \gamma ^{n}\ll \lambda ^{n}}$ , where ${\displaystyle \ll }$  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,
  $${\displaystyle \gamma ^{n}(A)=\sup\{\gamma ^{n}(K)\mid K\subseteq A,K{\text{ is compact}}\},}$$

   
  so Gaussian measure is a Radon measure;
• is not translation-invariant, but does satisfy the relation
  $${\displaystyle {\frac {\mathrm {d} (T_{h})_{*}(\gamma ^{n})}{\mathrm {d} \gamma ^{n}}}(x)=\exp \left(\langle h,x\rangle _{\mathbb {R} ^{n}}-{\frac {1}{2}}\|h\|_{\mathbb {R} ^{n}}^{2}\right),}$$

   
  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:
  $${\displaystyle Z\sim \operatorname {Normal} (\mu ,\sigma ^{2})\implies \mathbb {P} (Z\in A)=\gamma _{\mu ,\sigma ^{2}}^{n}(A).}$$

   

It can be shown that there is no analogue of Lebesgue measure on an infinite-dimensional vector space. Even so, it is possible to define Gaussian measures on infinite-dimensional spaces, the main example being the abstract Wiener space construction. A Borel measure γ on a separable Banach space E is said to be a non-degenerate (centered) Gaussian measure if, for every linear functional L ∈ E^∗ except L = 0, the push-forward measure L_∗(γ) is a non-degenerate (centered) Gaussian measure on R in the sense defined above.

For example, classical Wiener measure on the space of continuous paths is a Gaussian measure.

• Bogachev, Vladimir (1998). Gaussian Measures. American Mathematical Society. ISBN 978-1470418694.
• Stroock, Daniel (2010). Probability Theory: An Analytic View. Cambridge University Press. ISBN 978-0521132503.

• Besov measure – generalization of the Gaussian measure using the Besov normPages displaying wikidata descriptions as a fallback - a generalisation of Gaussian measure
• Cameron–Martin theorem – Theorem defining translation of Gaussian measures (Wiener measures) on Hilbert spaces.
• Covariance operator – Operator in probability theory
• Feldman–Hájek theorem – Theory in probability theory