Prerequisites

Basic measure theory, $\sigma$-fields, Lebesgue measure

Summary

This part takes concepts from measure theory to define probability measure and probability space. Then the Identitification Lemma for Probability is introduced to judge two probability measures are equal.

Probability Space

Definition 1   A set function ¹ ${\mathbb{P}}$ on a $\sigma$-field $\mathcal{F}$ is a probability measure if it satisfies the following conditions:

1. $0 \leq {\mathbb{P}}(A) \leq 1$ for $A \in \mathcal{F}$.
2. ${\mathbb{P}}(\emptyset) = 0, {\mathbb{P}}(\Omega) =1$.
3. If $A_i \in \mathcal{F}$ is a countable sequence of disjoint sets, then ${\mathbb{P}}(\bigcup_i A_i)=\sum_i {\mathbb{P}}(A_i)$.

If $\mathcal{F}$ is a $\sigma$-field, then the triple $(\Omega,\mathcal{F},{\mathbb{P}})$ is called a probability measure space or simply a probability space. The countable additivity of the probability measure gives rise to the following properties that are stated in a theorem.

Theorem 2   Let ${\mathbb{P}}$ be a probability measure on a field $\mathcal{F}$.

1. Continuity from below: If $A_n$ and $A$ lie in $\mathcal{F}$ and $A_n\uparrow A$, then ${\mathbb{P}}(A_n) \uparrow {\mathbb{P}}(A)$.
2. Continuity from above: If $A_n$ and $A$ lie in $\mathcal{F}$ and $A_n\downarrow A$, then ${\mathbb{P}}(A_n) \downarrow {\mathbb{P}}(A)$.
3. Countable subadditivity: If $A_1, A_2...$ and $\bigcup_{k=1}^{\infty} A_k$ lie in $\mathcal{F}$, then

   $${\mathbb{P}}\left(\bigcup_{k=1}^{\infty} A_k\right) \leq \sum_{k=1}^{\infty}{\mathbb{P}}(A_k).$$ (1)

   
   

Example 3   Let $\mathcal{R}=$ the Borel sets = the smallest $\sigma$-field containing the open sets. The probability space on a unit interval is then defined as $(\Omega,\mathcal{F},{\mathbb{P}})$, where $\Omega = (0,1)$, $\mathcal{F} = \{A \cap (0,1): A \in \mathcal{R}\}$ and ${\mathbb{P}}(B) = \lambda(B)$ for $B\in \mathcal{F}$. Here $\lambda$ is the Lebesgue measure restricted to the Borel subsets of (0,1).

Identitification Lemma for Probabilities

Now we will present a key result that would help us to extend the results we have on a field $\mathcal{A}$ to the $\sigma$-field generated by $\mathcal{A}$. This is stated as the Identitification Lemma for Probabilities:

Lemma 4 (Identification Lemma for Probabilities)   Let $P$ and $Q$ be two probability measures on $\sigma(\mathcal{A})$ where $\mathcal{A}$ is closed under intersections. If $P(A) = Q(A)$ for $A \in \mathcal{A}$, then $P(A)=Q(A)$ for all $A \in \sigma(\mathcal{A})$.

Before we prove this lemma, let's state some basic tools from measure theory that are very useful.

Definition 5   A collection of subsets $\mathcal{D}$ of set $\Omega$ is called a $\lambda$-system if

1. $\Omega \in \mathcal{D}$
2. If $A \in B$ and $B \in \mathcal{D}$, $A \subset B \Rightarrow B-A \in \mathcal{D}$
3. If $A_n \in \mathcal{D}$ and $A_n \uparrow A \Rightarrow A \in \mathcal{D}$

Theorem 6 (Dynkin's $\pi$-$\lambda$ Theorem)   Suppose $\mathcal{A}$ is a collection of sets closed under $\cap$ (a $\pi$-system). If $\mathcal{D}$ is a $\lambda$-system with $\mathcal{A} \subset \mathcal{D}$, then $\sigma(\mathcal{A}) \subset \mathcal{D}$.

With this theorem let's try to prove the identification lemma on probabilities.

Proof. (of Lemma ) Consider $\mathcal{D} = \{ A \in \sigma(\mathcal{A}): P(A)=Q(A)\}$. Let's check that $\mathcal{D}$ is a $\lambda$-system.

1. $\Omega \in \mathcal{D}$ because $P(\Omega)=Q(\Omega)=1$.
2. Let $A \in D , B \in \mathcal{D}$. Then $A \subset B$ implies $B-A \in \mathcal{D}$. This is because $P(B)=Q(B) \Rightarrow P(B-A) + P(A) = Q(B-A)+ Q(A) \Rightarrow P(B-A)=Q(B-A)$.
3. $A_n \in \mathcal{D}$ and $A_n \uparrow A$ implies $A \in \mathcal{D}$. This is because $P(A_n) = Q(A_n) \Rrightarrow P(A)=Q(A)$ using theorem .

Now directly applying Dynkins $\pi-\lambda$ Theorem, we get $\mathcal{A} \subset \mathcal{D}$ implies $\sigma(\mathcal{A}) \subset \mathcal{D}$. $\qedsymbol$

References

Durrett, Section 1.1

Footnotes

...http://planetmath.org/?op=getobj&from=objects&id=360 ¹

A set function is a real-valued function defined on some class of subsets of $\Omega$.