A quasi-invariance theorem for measures on Banach spaces

Abstract

We show that for a measure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma"> <mml:semantics> <mml:mi> γ </mml:mi> <mml:annotation encoding="application/x-tex">\gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on a Banach space directional differentiability implies quasi-translation invariance. This result is shown to imply the Cameron-Martin theorem. A second application is given in which <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma"> <mml:semantics> <mml:mi> γ </mml:mi> <mml:annotation encoding="application/x-tex">\gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the image of a Gaussian measure under a suitably regular map.