Sierpiński's theorem on non-atomic measures

Remarks

The first step shows that a set of positive finite measure contains measurable subsets of arbitrarily small positive measure. This uses the absence of atoms to split a set again and again into pieces of positive measure, keeping the smaller piece. The second step builds $B$ by adding, at each stage, a subset of nearly the largest size that keeps the total at most $b$. Continuity of measure then shows that the limit set has measure exactly $b$.

For Lebesgue measure on $[0, 1]$ the conclusion is easy: take $B = [0, b]$. The theorem matters for abstract non-atomic measures, such as the product measure of fair coin tosses on $\set{0, 1}^{\N}$.

The hypothesis that $\mu$ is non-atomic is needed. For the Dirac measure $\delta_x$ and $A = \set{x}$, every measurable subset of $A$ has measure $0$ or $1$, so no subset has measure $1/2$.

Going deeper

For a student

As a corollary, a finite non-atomic measure takes every value in $[0, \mu(X)]$, so its range is an interval. The Lyapunov convexity theorem extends this to finite non-atomic vector measures with values in $\R^n$: their range is compact and convex. The theorem is used in probability to build events of any prescribed probability, and in economics and game theory for fair division of non-atomic resources.