On some methods of extending invariant and quasi-invariant measures

Admissible vector fields and quasi-invariant measures∗

Quasi-invariant measures on the path space of a diffusion

The author has previously constructed a class of admissible vector fields on the path space of an elliptic diffusion process x taking values in a closed compact manifold. In this Note the existence of flows for this class of vector fields is established and it is shown that the law of x is quasi-invariant under these flows. Résumé L’auteur a précédemment construit une classe de champs de vecteu...

Nonexistence of Quasi-invariant Measures on Infinite-dimensional Linear Spaces1

It was shown by V. N. Sudakov [7] that a locally convex topological linear space with a nontrivial quasi-invariant <r-finite measure on its weakly measurable sets must be finite-dimensional. Sudakov's ingenious proof uses a theorem of Ulam [6, Footnote 3]. In some unpublished notes [8], Y. Umemara attempted to prove the same theorem, by utilizing Weil's "converse to the existence of Haar measur...

Extending Quasi-invariant Measures by Using Subgroups of a given Group

A method of extending σ-finite quasi-invariant measures given on an uncountable group, by using a certain family of its subgroups, is investigated. 2000 Mathematics Subject Classification: 28A05, 28D05.

On Invariant Measures

Given a measurable transformation on a measure space one can ask whether or not there is an equivalent measure that is invariant under the transformation. This problem is discussed very thoroughly in Halmos' Lectures on ergodic theory, pp. 81-90, 97. The first result along these lines is due to E. Hopf who obtained necessary and sufficient conditions for the existence of a finite invariant meas...