---
tag: J76687HP2E
kind: statement
role: theorem
name: Sierpiński's theorem on non-atomic measures
msc: [28A12]
uses: [P15VQS5DJ9, SV96KN4WR9, 0ACRGYNHSR]
about: [SV96KN4WR9]
sources:
  - url: https://en.wikipedia.org/wiki/Atom_(measure_theory)
---

## Statement

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]] such that $\mu$ is [[non-atomic|SV96KN4WR9]].
Let $A \in \Sigma$ with $\mu(A) < \infty$, and let $b$ be a real number with $0 \le b \le \mu(A)$.
Then there is a set $B \in \Sigma$ with $B \subseteq A$ and $\mu(B) = b$.

## 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|0ACRGYNHSR]] $\delta_x$ and $A = \set{x}$, every measurable subset of $A$ has measure $0$ or $1$, so no subset has measure $1/2$.

## Explanations

### L1 human

A measure without lumps can be cut into a piece of any size one likes.
If a set has measure $5$, one can find a part of it with measure exactly $\pi$, or $0.3$, or any number between $0$ and $5$.
Lumpy measures, such as point masses, cannot be cut like this.

### L2 human

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.