## The Wiener sphere and Wiener measure.(English)Zbl 0788.28008

Using Robinson styled nonstandard analysis and the Loeb measure, the authors define a uniform probability $$\mu_ L$$ on the infinite- dimensional sphere of Poincaré, Wiener and Lévy. Then they construct a Wiener measure from it. This gives a certain rigor to the informal discussion by McKean. The authors then provide an elementary proof of a weak convergence result and also study the infinite product of Gaussian measures. They investigate transformations of the sphere induced by shifts and the associated transformations of $$\mu_ L$$. Among other results, the Cameron-Martin density is derived as a Jacobian.

### MSC:

 28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) 28E05 Nonstandard measure theory 03H05 Nonstandard models in mathematics 60J65 Brownian motion 60H05 Stochastic integrals
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