Lyapunov theorem for $q$-concave Banach spaces

Mon, 28/10/2013 - 14:00

Let $X$ be Banach space, $(\Omega,\Sigma)$ is a measure space, where $\Omega$
is a set and $\Sigma$ is a $\sigma$-algebra of subsets of $\Omega.$ If $m:\Sigma\rightarrow X$ is
a $\sigma$-additive $X$-valued measure, then the range of $m$ is the set $m(\Sigma)=\{m(A):
\ A\in\Sigma.\}$ The measure $m$ is {\it non-atomic} if for every set $A\in\Sigma$ with $m(A)>0,$
there exist $B\subset A,B\in\Sigma$ such that $m(B)\neq0$ and $m(A \backslash B)\neq0.$
$X$-valued measure we will call {\it Lyapunov measure} if the closure of its range is convex.
And Banach space $X$ is {\it Lyapunov space} if every $X$-valued non-atomic measure is Lyapunov.

Theorem. Let X be Banach space with unconditional basis, q-concave, $q<\infty$, and which doesn't contain isomorphic copy of $l_2.$
Then X is Lyapunov space.