$\pi$–$\lambda$ Theorem

Monotone Class Lemma

Statement

Slogan version

$$\sigma = \pi + \lambda$$

$\sigma$ $\pi$ $\lambda$
“sum” “product” “limit”
universe nonempty universe
complement complement
countable union finite intersection disjoint countable union

A $\sigma$-algebra is a $\pi$-system and a $\lambda$-system, and vice versa.

Wiki version

A $\lambda$-system is a synonym of a Dykin system.

$$\mathcal{P} \subseteq \mathcal{D} \Longrightarrow \sigma(\mathcal{P}) \subseteq \mathcal{D}$$

Given a $\pi$-system contained in a $\lambda$-system. Then the $\sigma$-algebra generated by the $\pi$-system is also contained in the $\lambda$-system.

Wiki provided two equivalent definitions of a $\lambda$-system. Both of them involve a sequence of sets $(A_n)_n$ in $\mathcal{D}$. The differences are:

  1. $(A_n)_n$ is incresing
  2. $A_n$’s are pairwise disjoint

Their target is $\bigcup_n A_n \in \mathcal{D}$. Actually, the trick in #3 below shows that these two definitions mean the same thing.

Equivalence

$(S) \Rightarrow (W)$

Given $P \in \mathcal{P} \subseteq \mathcal{D}$.

  1. universe: $\Omega \in \mathcal{D}$
  2. stability under complement: $P^\complement \in \mathcal{D}$
  3. stability under countable union:

    Given $P_k \in \mathcal{D}$, $k \in \lbrace 1, \ldots,n \rbrace$. To make some pairwise disjoint sets $P^\prime_n$ so that $\bigcup_n P_n = \bigcup_n P^\prime_n$, set $P^\prime_n = P_n \setminus \left( \bigcup_{k=1}^{n-1} P_k \right)$ (i.e. $n$-th new contribution to the existing union $\bigcup_{k=1}^{n-1} P_k$). Verify that $P^\prime_n \in \mathcal{D}$ for all $n \in \N^*$.

I find #3 a bit tricky.

$(W) \Rightarrow (S)$

A $\sigma$-algebra is always a $\pi$-system and a $\lambda$-system.

Given a $\pi$-system $\mathcal{D}$ that is also a $\lambda$-system. Apply $(W)$ taking $\mathcal{P} = \mathcal{D}$. We arrive at $\sigma(\mathcal{D}) = \mathcal{D}$.

Examples

  1. $\pi$-system:
    • nested intervals: $\mathcal{\mathcal{P}}_1 = \lbrace (-\infty,x) \mid x \in \R \rbrace$
    • singletons: $\mathcal{\mathcal{P}}_2 = \lbrace \lbrace x \rbrace \in \Omega \rbrace \cup \lbrace \varnothing \rbrace$
    • rectangles: $\mathcal{\mathcal{P}}_3 = \lbrace \mathcal{P}_1 \times \mathcal{P}_2 \mid \mathcal{P}_1,\mathcal{P}_2 \in \mathcal{\mathcal{P}}(\Omega) \rbrace$
  2. $\lambda$-system:
    • to be continued…

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