Lyapunov's convexity theorem

Remarks

A short proof considers the set $K$ of measurable $g \colon X \to [0, 1]$, compact in the weak-$*$ topology of $L^\infty(\abs{\mu_1} + \dots + \abs{\mu_n})$. The map $g \mapsto \int g \, d\mu$ is linear and continuous, so its image is compact and convex, and it contains the range of $\mu$. For a point $v$ of the image, the preimage of $v$ in $K$ is compact and convex and has an extreme point by the Krein–Milman theorem. Non-atomicity forces such an extreme point to be an indicator function $\mathbf{1}_E$, so $v = \mu(E)$.

Example: for $n = 1$ and Lebesgue measure on $[0, 1]$, the range is $[0, 1]$, because $\lambda([0, t]) = t$.

Without non-atomicity the result fails: the Dirac measure $\delta_0$ has range $\{0, 1\}$, which is not convex. The result also fails in infinite dimensions. The vector measure $E \mapsto \mathbf{1}_E$ from the Borel sets of $[0, 1]$ to $L^2[0, 1]$ is non-atomic, but $\frac12 \mathbf{1}_{[0,1]}$ lies between $0$ and $\mathbf{1}_{[0,1]}$ and is not an indicator function.

Going deeper

For a student

The theorem is the basis of fair division results, such as the existence of exact divisions of a cake among several players. In control theory it gives the bang-bang principle: the set of states reachable with measurable controls is already reached with controls taking only extreme values. In economics it shows that non-atomic economies have convex aggregate behaviour.