I've recently been reading Billingsley's Probability and Measure Theory. I wanted to write down some notes I've been taking. In a separate post I'll also include my solutions to some of the chapters since the book doesn't provide complete solutions for all problems. This will likely be a series. This will contain notes for Chapter 2 and solutions to some of the problems. In general, for each article, I'll provide both notes and solutions and update them as I read further into the section and/or complete more problems. As a side note, since I'm trying to summarize important facts, I will not always include proofs for everything (though this might change as I update an article). I'll try to include updates/edits at the top of the article.

Probability Measures and Fields

A probability triple is a tuple $(\Omega, \mathcal{F}, P)$ where $\Omega$ is a set, $\mathcal{F}$ is a field, and $P$ is a set function from $\mathcal{F}$ into $\mathbb{R}$. A field $\mathcal{F}$ is a collection (or "class") of sets that satisfies the following:

1. $\Omega \in \mathcal{F}$
2. If $A \in \mathcal{F}$, then $A^c \in \mathcal{F}$
3. Let $A_1, A_2, ..., A_N \in \mathcal{F}$ be a finite collection of sets. Then, $\bigcup_{n=1}^N A_n \in \mathcal{F}$.

The set function $P$ is a probability measure that satisfies the following:

1. $0 \le P(A) \le 1$ for $A \in \mathcal{F}$
2. $P(\emptyset) = 0$, $P(\Omega) = 1$
3. If $A_1, A_2, ..., \text{etc}$ is a disjoint sequence of $\mathcal{F}$-sets and if $\bigcup_{k=1}^N A_k$, then $P(\cup_{k=1}^\infty A_k) = \sum_{k=1}^\infty P(A_k)$. That is, $P$ is countably additive.

This necessarily implies that $P$ is finitely additive as well, since we can take the first $N$ sets as $A_1, ..., A_N$, and then let $A_k = \emptyset$ for $k > N$.

σ-Fields

A $\sigma$ -field is a field that satisfies the following conditions:

1. $\Omega \in \mathcal{F}$
2. If $A \in \mathcal{F}$, then $A^c \in \mathcal{F}$.
3. Let $\{A_n\}_{n=1}^\infty$ is a sequence of $\mathcal{F}$-sets. Then, $\bigcup_{n=1}^\infty A_n \in \mathcal{F}$

In other words, the primary difference between a $\sigma$ -field and an ordinary field is that a $\sigma$ -field is closed under countable unions whereas ordinary fields are only closed under finite unions.

Let $\mathcal{A}$ be a class of subsets of $\Omega$. Then, the $\sigma$ -field generated by $\mathcal{A}$, denoted $\sigma(\mathcal{A})$, is the intersection of all $\sigma$ fields that contain $\mathcal{A}$. We have the two following facts:

1. $\sigma(\mathcal{A})$ is a $\sigma$ -field.
2. Let $\mathcal{A} \subset \mathcal{G}$, where $\mathcal{G}$ is a $\sigma$ -field. Then, $\sigma(\mathcal{A}) \subset \mathcal{G}$.

Important Properties of Probability Measure

1. Countable Subadditivity: If $\{A_n\}_{n=1}^\infty$ is a sequence of $\mathcal{F}$ -sets and $\cup_{k=1}^\infty A_k \in \mathcal{F}$ , then $P(\cup_{k=1}^\infty A_k) \le \sum_{k=1}^\infty P(A_k)$.
2. The Lebesgue measure $\lambda$ is a countably additive probability measure on the field $\mathcal{B}_0$. Here, the field $\mathcal{B}_0$ is the field consisting of finite disjoint unions of subintervals of $(0, 1] = \Omega$. If $A \in \mathcal{B}_0$, then we can write $A = \cup_{n=1}^N I_n$ where the $I_n$ are pairwise disjoint intervals in $(0, 1]$. Then, $\lambda(A) := \sum_{n=1}^N \lambda(I_n) = \sum_{n=1}^N |I_n|$.