---
tag: P0DV2TCH1N
kind: concept
name: Gaussian measure
msc: [28C20, 60G15, 60B11]
sources:
  - url: https://en.wikipedia.org/wiki/Gaussian_measure
same_as:
  wikidata: Q5527840
  wikipedia: Gaussian measure
---

## Summary

A Gaussian measure is a [[probability measure|059TJN0NZ1]] whose one-dimensional projections are all [[normal distributions|05AQE0QR0D]].
On $\R^n$ these are the laws of Gaussian random vectors.
On infinite-dimensional spaces, such as spaces of continuous paths, Gaussian measures are the main examples of well-behaved measures, and Wiener measure is one of them.
There is no translation-invariant analogue of Lebesgue measure in infinite dimensions, so Gaussian measures often take its place.

## Explanations

### L1 human

The standard Gaussian measure on the line is the bell curve, read as a measure: the mass of an interval is the area under the bell curve over it.
On the plane, the standard Gaussian measure is the product of two bell curves, and it looks like a round hill.
In general, a measure is Gaussian if every straight-line view of it, through a linear functional, is a bell curve or a single point.
For example, the law of the Brownian path on $[0, 1]$ is Gaussian, because each value of the path, and each linear combination of finitely many values, is normally distributed.