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4 — Outer Measure, Measurable Sets, Cantor Set, Non-Measurable Sets, σ-Finite Measures, Integration

Lecture 4 concludes the foundational measure theory part:

Outer measure and measurable sets

Cantor set and non-measurable sets

Borel vs Lebesgue σ-algebras

σ-finite measures

Beginning the theory of integration via simple functions

We assume we already have:

  • A set $\Omega$,
  • An algebra $\mathcal{A}$,
  • A premeasure $\mu$ defined on $\mathcal{A}$.

Goal: Extend $\mu$ to a measure on $\sigma(\mathcal{A})$ by using outer measure and Carathéodory measurability.

1. Outer Measure

Given $A \subseteq \Omega$, \(\mu^*(A) = \inf \left\{ \sum_{i=1}^{\infty} \mu(A_i) : A \subseteq \bigcup_{i=1}^\infty A_i,\; A_i \in \mathcal{A} \right\}.\)

Properties:

  1. Subadditivity
    \(\mu^*\!\left(\bigcup_{i=1}^\infty B_i\right) \le \sum_{i=1}^\infty \mu^*(B_i).\)

  2. Monotonicity
    If $B \subseteq D$, then
    \(\mu^*(B) \le \mu^*(D).\)

2. Carathéodory Measurability

A set $E \subseteq \Omega$ is measurable if: $$ \mu^(F) = \mu^(F \cap E)

  • \mu^*(F \cap E^c) \quad\forall F \subseteq \Omega. $$

Intuition (page 1 diagram):
A measurable set is one for which every set $F$ can be cleanly partitioned into “inside $E$” and “outside $E$” without losing measure.

Zero-measure sets are measurable

Claim:
If $\mu^*(E)=0$, then $E$ is measurable, and so is every subset of $E$.

Proof sketch:

  • $F \cap E \subseteq E$, hence
    \(0 \le \mu^*(F \cap E) \le \mu^*(E)=0.\)
  • Then
    \(\mu^*(F) = \mu^*(F \cap E) + \mu^*(F \cap E^c).\)

Thus all subsets of null sets are measurable.

3. σ-Algebra Generated and Lebesgue Measurability

Given the algebra $\mathcal{A}$ of finite unions of intervals $(a_i,b_i]$, we know:

\[\sigma(\mathcal{A}) = \text{Borel σ-algebra } \mathcal{B}.\]

Carathéodory produces:

  • A larger σ-algebra $\mathcal{A}^*$ of all Lebesgue measurable sets,
  • With $\mathcal{B} \subsetneq \mathcal{A}^*$.

4. Cantor Set (continued)

Let $F_n$ be the standard Cantor construction:

  • Remove the open middle third at each step,
  • $F = \bigcap_{n=1}^\infty F_n$.

Measure of Cantor set

\(\mu(F) = \lim_{n\to\infty} \left(\frac{2}{3}\right)^n = 0.\)

Cardinality of Cantor set

Using ternary expansions: \(F = \left\{ \sum_{i=1}^\infty \frac{\delta_i}{3^i} : \delta_i \in \{0,2\} \right\},\) which is in bijection with binary expansions of $[0,1]$.
Thus:

  • $\text{card}(F) = \text{card}(\mathbb{R})$,
  • But there are $2^{\text{card}(F)}$ subsets of $F$, which is strictly larger than $\mathbb{R}$.

Conclusion:
Most subsets of the Cantor set are not Lebesgue measurable.
The Lebesgue σ-algebra is too small to contain them.

5. Borel σ-Algebra vs Lebesgue σ-Algebra

  • Borel σ-algebra $\mathcal{B}$ = σ-algebra generated by open sets.
  • Lebesgue σ-algebra $\mathcal{A}^*$ includes all Borel sets plus all subsets of Lebesgue null sets.

Thus: \(\mathcal{B} \subsetneq \mathcal{A}^*.\)

Borel sets have “nicely definable structure,” but Lebesgue sets allow measurability of far more pathological sets.

6. Existence of Non-Measurable Sets (using Axiom of Choice)

The lecture constructs (page 1–2) a Vitali set inside $[0,1]$:

Consider equivalence relation: \(x \sim y \iff x - y \in \mathbb{Q}.\)

Pick exactly one representative from each equivalence class.
Call the set $B$.

Then:

  • The sets $B + q$ (shifts by rational numbers) are disjoint or nearly so.
  • \[\bigcup_{q \in \mathbb{Q} \cap [-1,1]} (B + q) \supseteq [0,1] \subseteq [-1,2].\]
  • If $B$ were measurable and translation-invariant, we would get contradictions such as
    \(\mu([0,1]) = \sum_{q} \mu(B), \qquad \mu([-1,2]) = 3.\)

Therefore:

Conclusion: Vitali sets are not Lebesgue measurable.

7. σ-Finite Measures

A measure space $(\Omega, \mathcal{F}, \mu)$ is σ-finite if:

\[\exists E_n \uparrow \Omega \quad \text{such that } \mu(E_n) < \infty.\]

Example: \(\lambda([-n,n]) = 2n < \infty, \quad \bigcup_{n=1}^\infty [-n,n] = \mathbb{R}.\)

Lebesgue measure on $\mathbb{R}$ is σ-finite even though $\lambda(\mathbb{R}) = \infty$.

σ-finiteness is essential for defining integration properly.

8. Measurable Functions

Given $(\Omega, \mathcal{F})$ and $(\mathbb{R}, \mathcal{B})$:

A function $f: \Omega \to \mathbb{R}$ is $\mathcal{F}$-$\mathcal{B}$ measurable if: \(f^{-1}(A) \in \mathcal{F}, \quad \forall A \in \mathcal{B}.\)

This is how random variables will be defined once probability is added.

9. Integration of Simple Functions

Start with a simple function: \(\varphi = \sum_{i=1}^n a_i \mathbf{1}_{A_i},\) where:

  • $A_i$ are measurable,
  • $A_i$ are disjoint,
  • $\mu(A_i)<\infty$.

Define the integral: \(\int_\Omega \varphi \, d\mu = \sum_{i=1}^n a_i \mu(A_i).\)

10. Properties of Integration (for simple functions)

Let $\varphi,\psi$ be simple functions, $a\in\mathbb{R}$:

  1. Non-negativity:
    If $\varphi \ge 0$, then $\int \varphi\, d\mu \ge 0$.

  2. Homogeneity:
    \(\int a\varphi\, d\mu = a \int \varphi\, d\mu.\)

  3. Additivity:
    \(\int (\varphi + \psi)\, d\mu = \int \varphi\, d\mu + \int \psi\, d\mu.\)

These are the algebraic foundations for Lebesgue integration.

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