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In Probability Theory: Separating Classes 📂Probability Theory

In Probability Theory: Separating Classes

Theorem

A Separating Class in a measurable space (S,B(S))(S, \mathcal{B}(S)) defined for two probabilities PP, QQ is said to satisfy the following C\mathcal{C}. P(A)=Q(A),∀A∈C  ⟹  P(A)=Q(A),∀A∈B(S) 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 C\mathcal{C} if σ(C)=B(S)\sigma (\mathcal{C}) = \mathcal{B}(S) and for all A∈CA \in \mathcal{C} it holds that P(A)=Q(A)P(A) = Q(A), then C\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. P\mathcal{P} that satisfies the following is called a π\pi-system. A,B∈P  ⟹  A∩B∈P A, B \in \mathcal{P} \implies A \cap B \in \mathcal{P}
  2. L\mathcal{L} that satisfies the following conditions is called a λ\lambda-system.
  • (i): ∅∈L\emptyset \in \mathcal{L}
  • (ii) A∈L  ⟹  Ac∈LA \in \mathcal{L} \implies A^{c} \in \mathcal{L}
  • (iii) For all i≠ji \ne j when Ai∩Aj=∅\displaystyle A_{i} \cap A_{j} = \emptyset {An}n∈N⊂L  ⟹  ⋃n∈NAn∈L\displaystyle \left\{ A_{n} \right\}_{n \in \mathbb{N}} \subset \mathcal{L} \implies \bigcup_{n \in \mathbb{N}} A_{n} \in \mathcal{L}

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

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

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

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

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

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