Let T be a non-empty set, A a u-algebra of subsets of T and u : .A -+ Rn a bounded,
countably additive measure. A set E E A is called an atom with respect to u if u(E)=/F 0
and, if F E A, FeE, then u(F) = u(E) or u(F) = 0; the measure u is atomic if there
exists at least one atom (with respect to u) in A. If no such atom (with respect to u)
exists in A, then u is called non-atomic.
In 1940 the Russian mathematician A. A. Lyapunov published the Convexity Theorem.
According to this theorem the range 'R.{u) of a bounded, finite-dimensional measure u
is compact and, in the non-atomic case, convex. Since 1940 much has been published
on different aspects of the range of a vector-measure. These aspects range from new
and shorter proofs of the Convexity Theorem and the usefulness of it in diverse fields,
to research about the geometrical characteristics of the range by using other familiar
theorems, like Krein-Milman and Radon-Nikodym.
In the survey at hand the Convexity Theorem in itself is studied. Applications in different
fields will be looked at as well as pieces about the history of the people and the ideas
involved in the development of the theorem.