From Probability Measures to a Deeper Question

Suppose you already know what a probability measure is and why random variables must be measurable. A natural next question is:

How do we actually construct such a measure?
Can we assign a measure to any set?

The answer is subtle: not every set can be measured. Carathéodory’s method explains how to build a measure and which sets are allowed.

Step 1. Start with Simple Measures

Begin by assigning length to half‑open intervals:

• For $(a,b] \subset \mathbb{R}$, set $\mu((a,b]) = b-a$.

This is finitely additive on the algebra generated by finite disjoint unions of such intervals.

Step 2. Outer Measure

To talk about arbitrary $E \subset \mathbb{R}$, define the outer measure

Because infima of nonempty subsets of $[0,\infty]$ always exist (possibly $=\infty$), the outer measure is defined for every set.

Step 3. Carathéodory-Measurable Sets

A set $E \subset \mathbb{R}$ is Carathéodory‑measurable if

Plainly put: splitting along $E$ does not break additivity of the outer measure. The collection of all such $E$ forms a $\sigma$‑algebra.

Step 4. Lebesgue Measure

Restricting $m^*$ to Carathéodory‑measurable sets yields a $\sigma$‑additive measure on that $\sigma$‑algebra. This is called the Lebesgue measure.

Step 5. Measurable vs Non‑Measurable: Two Canonical Examples

Cantor Set (Measurable yet “weird”)

Construct the Cantor set $C \subset [0,1]$ by repeatedly removing middle thirds:

1. Start with $[0,1]$; remove $(\tfrac13,\tfrac23)$.
2. From each remaining interval, remove its middle third; repeat ad infinitum.

Properties:

• $m(C)=0$ (the removed lengths sum to $1$),
• yet $C$ is uncountable (continuum cardinality),
• closed and compact,
• and Lebesgue‑measurable (it satisfies Carathéodory’s condition).

Interactive demo (build the Cantor set visually):

Vitali Set (Non‑Measurable, explained carefully)

Define an equivalence relation on $\mathbb{R}$ by $x \sim y \iff x-y \in \mathbb{Q}$. Using the axiom of choice, choose one representative from each equivalence class inside $[0,1]$; call the resulting set $V$ (a Vitali set).

Key facts behind non‑measurability (sketch):

1. For each rational $r \in \mathbb{Q} \cap [-1,1]$, consider the translate $V+r=\{v+r:v\in V\}$.
2. These translates are pairwise disjoint: if $(v_1+r_1)=(v_2+r_2)$ with $v_i\in V$ and $r_i\in\mathbb{Q}$, then $v_1-v_2=r_2-r_1\in\mathbb{Q}$; by construction $V$ picked one representative per rational class, hence $v_1=v_2$ and $r_1=r_2$.
3. We have
   $$[0,1]\ \subset\ \bigcup_{r\in\mathbb{Q}\cap[-1,1]} (V+r)\ \subset\ [-1,2].$$
4. If $V$ were Lebesgue‑measurable with measure $\alpha=m(V)$, then by translation invariance and countable additivity,
   $$\sum_{r\in\mathbb{Q}\cap[-1,1]} m(V+r)\ =\ \sum_{r} \alpha.$$
   • If $\alpha>0$, the sum diverges to $\infty$, contradicting $m([-1,2])=3$.
   • If $\alpha=0$, the sum is $0$, contradicting that the union covers $[0,1]$, which has measure $1$.

Hence $V$ cannot be measurable: Lebesgue measure cannot be consistently assigned to it. The construction crucially uses the axiom of choice.

Summary

• Outer measure $m^*$ exists for every set (infimum always exists).
• Carathéodory‑measurable sets are exactly those for which splitting preserves additivity.
• Restricting $m^*$ to them gives the Lebesgue measure.
• Not every set is measurable:
  • Cantor set: measurable, $m(C)=0$, uncountable.
  • Vitali set: non‑measurable; measurability would contradict translation invariance + countable additivity.