Fields

1. Core Concept Overview & Intuition

The theory of fields is the structural backbone of probability and measure theory. The central idea is that we rarely assign probabilities to every conceivable subset of a sample space; instead we work with a carefully chosen collection of subsets that is rich enough to support the operations we need (complementation, finite intersections, unions) yet well-behaved enough to avoid logical pathologies. A field is precisely such a collection: a non-empty class of subsets of a universal set $U$ that is closed under complementation and finite intersections. From these two closure properties, all the familiar set operations — unions, differences, the empty set, the universal set — follow automatically.

The intuition is that a field encodes the information available to an observer. If an event $A$ is "observable" (belongs to the field), then so is its negation $A^{c}$; if two events are observable, so is their simultaneous occurrence $A\cap B$. DeMorgan's rule then guarantees that "either $A$ or $B$" is also observable. Strengthening finite closure to countable closure yields a $\sigma$-field, the natural home for limiting operations such as $\lim A_n$. A monotone field closes only under monotone limits, and a key theorem shows that monotone fields and $\sigma$-fields coincide — a bridge between the finite and the countable.

On top of this set-theoretic scaffolding sit functions. A function $X:U\to U'$ assigns to each $w\in U$ a value $X(w)=w'$. When the codomain is the real line we obtain real-valued functions; when the domain consists of sets we obtain set functions; the inverse image $X^{-1}(D)$ is itself a set function. Requiring $X^{-1}(B)\in\mathcal{F}$ for every Borel set $B$ produces measurable functions, and pairing $U$ with a $\sigma$-field $\mathcal{F}$ produces a measurable space $(U,\mathcal{F})$. Adding a countably additive set function $\mu$ with $\mu(\varnothing)=0$ and $\mu(A)\ge 0$ produces a measure space $(U,\mathcal{F},\mu)$.

Probability is the special case where the measure of the whole space equals one. The axiomatic approach packages this as a triple $(S,\mathcal{F},P)$ with $0\le P(A)\le 1$, $P(S)=1$, and countable additivity over pairwise disjoint events. Random variables and their distribution functions $F_X(x)=P[X\le x]$ then translate probabilistic questions into real analysis. Throughout, the guiding principle is closure: start with the operations you need, and build the smallest collection of sets (or the largest structure) that supports them.

2. Mathematical Formalization & Core Definitions

2.1 Sequences and Limits (Prerequisite Recall)

A sequence of sets ${B_n}$ is increasing if $B_1\subseteq B_2\subseteq\cdots$ and decreasing if $B_1\supseteq B_2\supseteq\cdots$. For an increasing sequence the limit is the union, and for a decreasing sequence the limit is the intersection.

> Example (increasing sequence of intervals). Let

$$
B_n=\left{w:2+\frac{1}{n}<w<6-\frac{1}{n}\right}.
$$

> Then $B_1={w:5<w<5}=\varnothing$, $B_2={w:3.5<w<5.5}$, $B_3={w:3<w<5.67}$, and so on. The sequence is increasing with limit $B={w:2<w<6}$.

This example illustrates that a sequence may begin with the empty set and still converge to a non-trivial limit; the limit operation is what motivates the countable closure requirements below.

2.2 Fields

> Definition (Field). A field is a non-empty class $\mathcal{F}$ of subsets of $U$ which is (1) closed under complementation, and (2) closed under finite intersections.

Let $\mathcal{F}$ be a field. Then:

1. if $A\in\mathcal{F}$ then $A^{c}\in\mathcal{F}$;
2. if $A\in\mathcal{F}$ and $B\in\mathcal{F}$ then $A\cap B\in\mathcal{F}$.

2.3 DeMorgan's Rule

> DeMorgan's Rule. If $A,B\in\mathcal{F}$ then $A\cup B\in\mathcal{F}$.

Proof. Let $A,B\in\mathcal{F}$. By rule 1, $A^{c}\in\mathcal{F}$ and $B^{c}\in\mathcal{F}$, and by rule 2 applied to $A^{c}$ and $B^{c}$ we get $(A^{c}\cap B^{c})\in\mathcal{F}$; hence by rule 1 again $(A^{c}\cap B^{c})^{c}\in\mathcal{F}$. But by DeMorgan's law $(A^{c}\cap B^{c})^{c}=A\cup B$, so that $A\cup B\in\mathcal{F}$. $\blacksquare$

2.4 The Empty Set and the Universal Set

> Result 1. The empty set $\varnothing$ is a member of every field.

Proof. If $A\in\mathcal{F}$ then $A^{c}\in\mathcal{F}$ by rule 1, so by rule 2, $A\cap A^{c}=\varnothing\in\mathcal{F}$. $\blacksquare$

> Result 2. The universal set $U$ is a member of every field.

Proof. If $A\in\mathcal{F}$ then $A^{c}\in\mathcal{F}$ by rule 1, so by DeMorgan's rule, $A\cup A^{c}=U\in\mathcal{F}$. $\blacksquare$

2.5 Minimal Field, $\sigma$-Field, Borel Field, Monotone Field

> Definition (Minimal field). Let $A$ be a set and $U$ the universal set. The minimal field of the set $A$ is the smallest field containing $A$, namely

$$
\mathcal{F}_A={\varnothing,\,A,\,A^{c},\,U}.
$$

> This is the field said to be generated by $A$.

> Definition ($\sigma$-field). A $\sigma$-field is a field which is (1) closed under complementation, and (2) closed under countable intersections and countable unions.

> Definition (Borel field). A Borel field is a field which contains the intervals of the real line

$$
(-\infty,a),\quad(-\infty,a],\quad(b,\infty),\quad[b,\infty),\quad(a,b),\quad[a,b),\quad(a,b],\quad[a,b],
$$

> and is denoted by $\mathcal{B}$. It is the $\sigma$-field generated by the class of all such intervals.

> Definition (Monotone field). A field $\mathcal{F}$ is a monotone field if it is closed under monotone operations: if ${A_n}$ is a monotone sequence in $\mathcal{F}$ and $\lim A_n\in\mathcal{F}$, then $\mathcal{F}$ is a monotone field.

2.6 Functions

Let $U$ be a universal set and let $w\in U$. Let $X(w)=w'$ be the value associated with $w$ when $w$ is operated on by $X$. If $w'\in U'$ for every $w\in U$, then $X$ is a function, or mapping, that maps $U$ into $U'$, written $X:U\to U'$. Here $U$ is the domain and $U'$ the range (codomain) of $X$. Functions are classified as:

1. one-to-one — each $w$ gives a different $w'$;
2. one-to-many — one $w$ is associated with several values;
3. many-to-one — several $w$ give the same $w'$.

> Definition (Real-valued function). A function which maps into the real line, $X:U\to\mathbb{R}$.

> Definition (Set function). A function whose domain contains sets, that is, it operates on sets.

> Definition (Inverse function). An inverse function is a set function. Let $X:U\to U'$ and let $D\subset U'$. Then

$$
X^{-1}(D)={w:X(w)=w'\in D}\subset U.
$$

> Definition (Measurable function). Let $U$ be a universal set and let $\mathcal{F}$ be a $\sigma$-field over $U$, so that all sets of $\mathcal{F}$ are subsets of $U$. Let $X$ be a mapping of $U$ into the real line, $X:U\to\mathbb{R}$, and let $B\in\mathcal{B}$, where $\mathcal{B}$ is the Borel field. If

$$
X^{-1}(B)=A,\quad\text{where }A\in\mathcal{F},
$$

> then $X$ is a measurable function with respect to the $\sigma$-field $\mathcal{F}$.

> Definition (Borel function). A function is a Borel function if it maps the real line into the real line and is measurable with respect to $\mathcal{B}$.

> Definition (Measurable space). Let $X:U\to\mathbb{R}$ be such that for every $B\in\mathcal{B}$ the set $X^{-1}(B)=A$ lies in $\mathcal{F}$, $\mathcal{F}$ being a $\sigma$-field over $U$. Then the pair $(U,\mathcal{F})$ is called a measurable space.

> Definition (Measure). Let $X:U\to\mathbb{R}$ and let $(U,\mathcal{F})$ be a measurable space. Define a function $\mu$ on $\mathcal{F}$ such that (1) $\mu(\varnothing)=0$; (2) $\mu(A)\ge 0$ for all $A\in\mathcal{F}$; (3) $\mu$ is countably additive: if $A_1,A_2,\dots,A_n,\dots$ are pairwise disjoint sets of $\mathcal{F}$, then

$$
\mu\left(\bigcup_{i=1}^{\infty}A_i\right)=\sum_{i=1}^{\infty}\mu(A_i).
$$

> Then $\mu$ is called a measure on the measurable space $(U,\mathcal{F})$, and the triple $(U,\mathcal{F},\mu)$ is a measure space. If $\mu(U)<\infty$ then $\mu$ is a finite measure. If $\mu(U)=\infty$ but there exist $A_1,A_2,\dots$ with $\bigcup_{i=1}^{\infty}A_i=U$ and $\mu(A_i)<\infty$ for each $i$, then $\mu$ is a $\sigma$-finite measure.

2.7 Indicator and Simple Functions

> Definition (Indicator function). Let $U$ be a universal set and let $A\subset U$. The indicator function of the set $A$ is defined by

$$
I_A(w)=\begin{cases}1, & w\in A,\ 0, & w\in A^{c}.\end{cases}
$$

> Definition (Simple function). A simple function is a finite linear combination of indicator functions of sets which are not necessarily disjoint:

$$
X=C_A I_A+C_B I_B,
$$

> where $C_A$ and $C_B$ are constants.

> Definition (Partition). Let $U$ be a universal set and let $A_1,A_2,\dots,A_n,\dots$ be non-empty, pairwise disjoint subsets of $U$ such that

$$
\bigcup_{i}A_i=U.
$$

> Then the class ${A_i;\ i=1,2,\dots,n}$ is a partition of $U$.

2.8 Probability

Probability is a numerical measure of the chance of occurrence of events. Let $A$ be an event. Then:

1. $0\le P(A)\le 1$;
2. $P(\varnothing)=0$;
3. if ${A_i;\ i=1,2,\dots,n}$ is a partition of $U$, then

$$
P\left(\bigcup_{i=1}^{n}A_i\right)=\sum_{i=1}^{n}P(A_i).
$$

> Definition (Experiment). An activity whose outcome cannot be determined with certainty.

> Definition (Sample space). The set of all possible outcomes.

> Definition (Outcome). An element of the sample space.

For example, tossing a coin is an experiment and the outcomes are $H$ and $T$:

$$
S={H,T},\quad E_1={\varnothing},\quad E_2={H},\quad E_3={T},\quad E_4={H,T}.
$$

2.9 Random Variables and Distribution Functions

> Definition (Random variable). A random variable is a function which takes its values according to the outcomes of an experiment; for example, in two throws of a coin, $X=0,1,2$ if we are interested in the number of tails.

> Definition (Distribution function). Let $X$ be a random variable on the probability space $(S,\mathcal{F},P)$. For $x\in\mathbb{R}$ define

$$
F_X(x)=P[X\le x],
$$

> and for $a\le b$,

$$
P[a<X\le b]=F_X(b)-F_X(a).
$$

> Then $F_X(x)$ is called the distribution function of $X$.