Publication at Faculty of Mathematics and Physics |
2009
Abstract
Let $\phi:X\to Y$ be an affine continuous mapping of a compact convex set X onto a compact convex set Y. We show that the induced mapping $\phi#$ need not map maximal measures on X to maximal measures on Y even in case $\phi$ maps extreme points of X to extreme points of Y.
This disproves Théoréme 6 of [S. Teleman, Sur les mesures maximales, C.
R. Acad.
Sci. Paris Sér.
I Math. 318 (6) (1994) 525-528]. We prove the statement of Théoréme 6 under an additional assumption that extY is Lindelöf or Y is a simplex.
We also show that under either of these two conditions injectivity of $\phi$ on extX implies injectivity of $\phi#$ on maximal measures. A couple of examples illustrate the results.