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3 — π-systems, λ-systems, π–λ Theorem, Outer Measure, Lebesgue Measure

Lecture 3 is a major theoretical step: it introduces the π–λ theorem, then constructs outer measure and Lebesgue measurable sets, culminating in the extension of pre-measure → Lebesgue measure on the Borel σ-algebra.

We begin with a pre-measure $\mu$ on $\Omega = (0,1]$ defined on the algebra \(\mathcal{A} = \left\{ \bigcup_{i=1}^m (a_i, b_i] : a_i < b_i \right\}.\)

Goal:
Extend $\mu$ to a measure on the σ-algebra $\sigma(\mathcal{A})$, the Borel σ-algebra of $(0,1]$.

1. Borel σ-Algebra and Its Minimality

The Borel σ-algebra $\mathcal{B}$ is the smallest σ-algebra containing all open sets.
Since \((a,b) = \bigcup_{n=1}^\infty \left(a, b - \frac{1}{n}\right],\) we see that every open interval belongs to $\sigma(\mathcal{A})$.

Thus: \(\sigma(\mathcal{A}) = \text{Borel σ-algebra}.\)

Two basic questions:

  1. Existence of extension of $\mu$ to $\sigma(\mathcal{A})$: Yes.
  2. Uniqueness of extension: Yes.
    (Requires π–λ machinery.)

2. π-systems

A collection $\pi \subseteq 2^\Omega$ is a π-system if:

  • $\Omega \in \pi$,
  • If $A,B \in \pi$, then $A \cap B \in \pi$.

Examples:

  • Intervals of the form $(a,b]$.
  • Rectangles in $\mathbb{R}^d$.
  • Any generating set for a Borel σ-algebra.

π-systems capture finite intersection structure.

3. λ-systems (Dynkin systems)

A collection $\lambda \subseteq 2^\Omega$ is a λ-system if:

  1. $\Omega \in \lambda$.
  2. If $A,B \in \lambda$ with $A \subseteq B$, then
    \(B \setminus A \in \lambda.\)
  3. For disjoint $A_1, A_2, \dots \in \lambda$,
    \(A_1 \cup A_2 \cup \cdots \in \lambda.\)

Important: A λ-system is not necessarily closed under intersections, so it is not a σ-algebra.

The lecture’s diagram on page 2 of the PDF shows a λ-system containing sets
${\varnothing, \Omega, A, A^c, B, B^c}$,
but not $A \cap B$.

4. Making a λ-system into a σ-algebra

If a λ-system is also a π-system, then it is automatically a σ-algebra:

  • π-system gives closure under intersections,
  • λ-system gives closure under complements and countable unions.

Thus: \(\text{π-system} + \text{λ-system} = \text{σ-algebra}.\)

This motivates the central theorem.

5. The π–λ Theorem (Dynkin)

Theorem (π–λ theorem).
If $\pi$ is a π-system, then the smallest λ-system containing it is exactly the σ-algebra it generates: \(\lambda(\pi) = \sigma(\pi).\)

This theorem is extremely useful because λ-systems are easier to show closed under operations needed to prove measure uniqueness.

6. Application: Uniqueness of Measure Extension

Suppose $V_1$ and $V_2$ are measures on $(\Omega, \sigma(\pi))$ such that:

  • They agree on the π-system $\pi$,
  • $V_1(\Omega) = V_2(\Omega) < \infty$.

Define: \(\lambda = \{B \in \sigma(\pi) : V_1(B) = V_2(B)\}.\)

We check the λ-system properties.

(1) $\Omega \in \lambda$

Because $V_1(\Omega)=V_2(\Omega)$.

(2) Closure under differences

If $A,B \in \lambda$ and $B \subseteq A$, then: \(V_1(A \setminus B) = V_1(A) - V_1(B) = V_2(A) - V_2(B) = V_2(A \setminus B),\) so $A \setminus B \in \lambda$.

(3) Closure under increasing unions

If $A_n \uparrow A$ with all $A_n \in \lambda$, then: \(V_1(A_n) \uparrow V_1(A), \qquad V_2(A_n) \uparrow V_2(A),\) so $A \in \lambda$.

Thus $\lambda$ is a λ-system, and since it contains $\pi$, Dynkin’s theorem gives: \(\sigma(\pi) = \lambda(\pi) \subseteq \lambda.\)

Therefore $V_1 = V_2$ on all of $\sigma(\pi)$.
This proves uniqueness of measure extension.

7. Carathéodory Outer Measure

To prove existence of Lebesgue measure, we construct an outer measure $\mu^*$.

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

Interpretation:
Cover $A$ with countably many semi-open intervals and take the smallest total measure.

Properties of outer measure (page 3–4)

  1. $\mu^*(\varnothing)=0$.
  2. If $E \subseteq F$ then
    \(\mu^*(E) \le \mu^*(F).\)
  3. For any countable family $A = \bigcup_{i=1}^\infty A_i$,
    \(\mu^*(A) \le \sum_{i=1}^\infty \mu^*(A_i).\)

8. Carathéodory Measurable Sets

A set $E \subseteq \Omega$ is Carathéodory measurable if:

\[\mu^*(F) = \mu^*(F \cap E) + \mu^*\big(F \cap E^c\big) \quad\text{for all } F \subseteq \Omega.\]

Geometric intuition (seen in the page-4 diagram):
For any “test” set $F$, the measure of $F$ must split exactly into the measure of the part inside $E$ and the part outside.

Define: \(\mathcal{A}^* = \{E : E \text{ is measurable}\}.\)

9. Main Results of Carathéodory’s Theorem

  1. $\mathcal{A} \subseteq \mathcal{A}^*$
    (The original algebra’s sets are measurable.)

  2. $\mathcal{A}^*$ is a σ-algebra.

  3. $\mu^*$ restricted to $\mathcal{A}^*$ is a complete measure.

  4. If $\mu^*(E) = 0$, then $E \in \mathcal{A}^$
    (Outer measure is *complete    ).

  5. For our original pre-measure on semiopen intervals,
    \(\sigma(\mathcal{A}) = \mathcal{B} \subseteq \mathcal{A}^*,\) and the extension of $\mu$ to $\mathcal{B}$ is exactly Lebesgue measure.

Thus, we have:

  • Existence of Lebesgue measure,
  • Uniqueness from the π–λ theorem.
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