Abstract
One of the classical results of vector measure theory is
A. A. Lyapunov´s which states that the range of every countably
additive finite nonatomic vector measure valued in a finite
dimensional space is a compact and convex set. This result,
first obtained by A. A. Lyapunov in 1940 attracted the attention of
many mathematicians. It is known that the Lyapunov theorem is false
in the infinite-dimensional case. The aim of this thesis is to develop
infinite-dimensional generalizations of the Lyapunov theorem.
A Banach space is said to have the Lyapunov property if the closure
of the range of every countably additive finite nonatomic vector
measure valued in this space is a convex set.
The present work consists of three chapters. General information on
vector measures, different approaches to generalizations of the
Lyapunov convexity theorem and the three-space problem for the
Lyapunov property are presented in Chapter 1. In Chapter 2 the
notions of a Lyapunov tree and Lyapunov B- and C-convexity are
introduced. It is proved that Banach spaces having this property have
the Lyapunov property. In Chapter 3 we find out which of the known
Banach spaces have the Lyapunov property and which do not. Most of the
constructed examples fit into the scheme: if a Banach space has no
isomorphic copies of l2 then it has the Lyapunov property. However,
we present an example of a Banach space that does not fit into this
scheme.
One of the classical results of vector measure theory is
A. A. Lyapunov´s which states that the range of every countably
additive finite nonatomic vector measure valued in a finite
dimensional space is a compact and convex set. This result,
first obtained by A. A. Lyapunov in 1940 attracted the attention of
many mathematicians. It is known that the Lyapunov theorem is false
in the infinite-dimensional case. The aim of this thesis is to develop
infinite-dimensional generalizations of the Lyapunov theorem.
A Banach space is said to have the Lyapunov property if the closure
of the range of every countably additive finite nonatomic vector
measure valued in this space is a convex set.
The present work consists of three chapters. General information on
vector measures, different approaches to generalizations of the
Lyapunov convexity theorem and the three-space problem for the
Lyapunov property are presented in Chapter 1. In Chapter 2 the
notions of a Lyapunov tree and Lyapunov B- and C-convexity are
introduced. It is proved that Banach spaces having this property have
the Lyapunov property. In Chapter 3 we find out which of the known
Banach spaces have the Lyapunov property and which do not. Most of the
constructed examples fit into the scheme: if a Banach space has no
isomorphic copies of l2 then it has the Lyapunov property. However,
we present an example of a Banach space that does not fit into this
scheme. | 
