MEASURE PRESERVING FUNCTIONS ON LOCALLY COMPACT SPACES,
Abstract
The following theorem is proved: Let M and N be compact metric spaces with Borel measures u and v, respectively, such that every non-empty open set has positive measure. Let f : M N be a continuous function having the property; whenever both A and f(A) are Borel measurable, then u(A) = v(f(A)). Then there exists a subset F of M of the first category and of measure zero on which f is not one-to-one and such that f : M - F is a homeomophism. This result without the continuity assumption is then generalized to locally compact metric spaces. The conclusion is slightly weaker. Some examples are given to prove that other likely extensions are false in general. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Apr 01, 1961
- Accession Number
- AD0256667
Entities
People
- M. E. Mahowald
- P. T. Church
Organizations
- Syracuse University