Figure 2

Atomic measure. The random variable Y lands with equal probability in Y=5/2 and uniformly in the interval [0,1]. The measure$w_Y\left(E\right):=2\underset{E}{\int }\mathrm{ydY}$ is atomic because the set {1} has strictly positive probability and no set other than the empty set is strictly included in {1}. Theorem 2 does not apply. The range of w _Y (E) is the nonconvex union of the intervals [0,1/2] and [5/2,3].