Gaussian Measures On Infinite-dimensional Spaces
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.
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