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On heredity of strongly proximal actions

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*C. Robinson Edward Raja*

**Address.** Stat-Math Unit, Indian Statistical Institute, 8th Mile
Mysore Road, R. V. College Post, Bangalore - 560 059, India
**E-mail:** creraja@isibang.ac.in,
raja_robinson@hotmail.com

**Abstract.**

We prove that action of a semigroup $T$ on compact metric space $X$
by continuous

selfmaps is strongly proximal if and only if $T$ action on ${\mathcal
P}(X)$ is strongly

proximal. As a consequence we prove that affine actions on certain
compact convex

subsets of finite-dimensional vector spaces are strongly proximal if
and only if the action

is proximal.

**AMSclassification.** 37B05, 60B05.

**Keywords.** Proximal and strongly proximal actions, and probability
measures.