Let X be a Banach space and A ⊂ X an absolutely convex, closed, and bounded set. We give some sufficient and necessary conditions in order that A lies in the range of a measure valued in the bidual space X ∗ ∗ and having bounded variation. Among other results, we prove that X ∗ is a G. T.-space if and only if A lies inside the range of some X ∗ ∗ -valued measure with bounded variation whenever X A is isomorphic to a Hilbert space.