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The Radon–Nikodym derivative of the transformed Gaussian measure n(T) with respect to n
For , Shale defines a new measure on the Gaussian space via pullback on , and sets
Key properties (paper use):
Shale's restricted general linear group is
Let be a real Hilbert space. A normal distribution over in Segal's sense is a probability space carrying a real-linear map
such that the random variables are jointly centered Gaussian and
Thus is an isometric embedding of into the real Gaussian subspace of .
Let and be -finite positive measures on the same measurable space. If , the Radon–Nikodym derivative is the -almost-everywhere unique measurable function such that
for every measurable set .
Let be a vector space over a field . A quadratic form on is a function such that
for all , , and the polarization
is a bilinear form. The pair is called a quadratic space.