📂Probability Theory

In Probability Theory: Separating Classes

Theorem

A Separating Class in a measurable space $(S, \mathcal{B}(S))$ defined for two probabilities $P$, $Q$ is said to satisfy the following $\mathcal{C}$. $$ P(A) = Q(A), \forall A \in \mathcal{C} \implies P(A) = Q(A), \forall A \in \mathcal{B}(S) $$

Explanation

The existence of a separating class implies that to check if two measures are the same, one does not need to examine the entire measurable space but only a part of it. Intuitively, it seems unlikely for such a convenient class to simply exist, but according to the following theorem, it can be found under relatively easy conditions.

Theorem

For a π-system $\mathcal{C}$ if $\sigma (\mathcal{C}) = \mathcal{B}(S)$ and for all $A \in \mathcal{C}$ it holds that $P(A) = Q(A)$, then $\mathcal{C}$ is a separating class.

Usage

The condition of being a π-system is much more approachable compared to directly satisfying the definition of a separating class. If such a theorem exists, it will be easier to show the existence of a separating class, ultimately making it easier to prove that the two probabilities (measures) are equal.

Proof

To use the Dynkin π-λ theorem, the following definitions are introduced.

π-system and λ-system:

1. $\mathcal{P}$ that satisfies the following is called a $\pi$-system. $$ A, B \in \mathcal{P} \implies A \cap B \in \mathcal{P} $$
2. $\mathcal{L}$ that satisfies the following conditions is called a $\lambda$-system.

• (i): $\emptyset \in \mathcal{L}$
• (ii) $A \in \mathcal{L} \implies A^{c} \in \mathcal{L}$
• (iii) For all $i \ne j$ when $\displaystyle A_{i} \cap A_{j} = \emptyset$ $$\displaystyle \left\{ A_{n} \right\}_{n \in \mathbb{N}} \subset \mathcal{L} \implies \bigcup_{n \in \mathbb{N}} A_{n} \in \mathcal{L}$$

Defining $\mathcal{L} := \left\{ A \in \mathcal{B}(S) : P(A) = Q(A) \right\}$,

• (i): Since $P( \emptyset ) = Q (\emptyset ) = 0$, it holds that $\emptyset \in \mathcal{L}$.
• (ii): Since $P(A^{c}) = 1 - P(A) = 1 - Q(A) = Q(A^{c})$, it holds that $A \in \mathcal{L} \implies A^{c} \in \mathcal{L}$.
• (iii): As the probabilities $P$, $Q$ are measures, they satisfy $\displaystyle \left\{ A_{n} \right\}_{n \in \mathbb{N}} \subset \mathcal{L} \implies \bigcup_{n \in \mathbb{N}} A_{n} \in \mathcal{L}$. Therefore, $\mathcal{L}$ is a λ-system.

Since it was assumed that for all $A \in \mathcal{C}$, $P(A) = Q(A)$ was satisfied, therefore $\mathcal{C} \subset \mathcal{L}$ is true.

Dynkin’s π-λ theorem: If a π-system $\mathcal{P}$ is a subset of a λ-system $\mathcal{L}$, then there exists a σ-field $\sigma ( \mathcal{P} )$ satisfying $\mathcal{P} \subset \sigma ( \mathcal{P} ) \subset \mathcal{L}$.

Since $\mathcal{C}$ was a π-system by assumption, according to Dynkin’s π-λ theorem, there exists a $\sigma ( \mathcal{C}) = \mathcal{B}(S)$ satisfying $\mathcal{C} \subset \sigma ( \mathcal{C}) \subset \mathcal{L}$. Naturally, the definition of $\mathcal{L}$ implies $\mathcal{L} \subset \mathcal{B}(S)$, so $\mathcal{L} = \mathcal{B}(S)$, and that $A \in \mathcal{C}$ satisfying $P(A) = Q(A)$ also belongs to $\mathcal{B}(S)$. In other words, for all $A \in \mathcal{B}(S)$, $P(A) = Q(A)$ holds, and rephrased as a proposition, it is as follows. $$ P(A) = Q(A), \forall A \in \mathcal{C} \implies P(A) = Q(A), \forall A \in \mathcal{B}(S) $$

■