Figure 1

Nonatomic measure. The random variable X is uniformly distributed in$[0,1]\cup [2,3]$. The measure$w_X\left(E\right):=2\underset{E}{\int }\mathrm{xdX}$ is nonatomic because all sets of nonzero probability include a smaller set of nonzero probability. Lyapunov’s convexity theorem (Theorem 2) states that the measure range$\mathcal{W}:=\left\{w_X\right(E):E\in \mathbb{B}\}$ is convex. The range of w _X is the, indeed convex, interval [0,3].