Definitions

1. Core Concept Overview & Intuition

The unit Definitions fixes the vocabulary on which the whole of the course is built. Everything later — events, measures, random variables — is expressed in the language of sets, classes of sets, and functions between sets. The guiding idea is a strict separation of three levels of objects: individual elements, sets built from those elements, and classes whose members are themselves sets. Confusing these levels is the single most common source of error in the subject.

The notation used throughout is fixed as follows. $(U,\mathcal{F},P)$ denotes a probability space, where $U$ is the universal (sample) set, $\mathcal{F}$ is the $\sigma$-field of events, and $P$ is the probability measure. Further, $\mathcal{B}$ is the Borel $\sigma$-field of the real line, $\mathbb{R}$ is the real line, $\varnothing$ is the empty set, $A^{c}$ is the complement of $A$ in $U$, and $I_{A}$ is the indicator function of $A$.

Intuitively, a field is a collection of sets that is stable under the basic operations of complementation and finite intersection; a $\sigma$-field extends this stability to countable operations. A sequence of sets describes how a family of sets grows or shrinks, and its limit records the set that the family eventually fills or eventually leaves. A function assigns to each element of a domain a value in a codomain, and a set function is simply a function whose inputs are sets rather than points.

2. Mathematical Formalization & Core Definitions

2.1 Sets, Classes and Their Algebra

> Set. A set is a collection or group of objects with similar characteristics.

> Null (empty) set. A set with no objects inside it; denoted $\varnothing$.

The running examples are

$$
A={a,b,c},\qquad B={a,e},\qquad C={d},\qquad \varnothing={}.
$$

> Universal set. The set containing all possible objects.

For the examples above the universal set is $U={a,b,c,d,e}$.

> Class. A set whose objects are sets.

For instance, with $\tau={A,B,C}$ and $\alpha={D,E}$, the collection $\tau$ is a class whose elements are the sets $A$, $B$, $C$. The three levels are kept distinct as follows.

> Subset. Let $A$ be a set and let $B$ be another set. Then $A$ is a subset of $B$ if all the elements of $A$ are members of $B$, that is

$$
A\subset B\iff(w\in A\Rightarrow w\in B).
$$

> Equality. If all members of $A$ are members of $B$ and all members of $B$ are members of $A$, then $A$ and $B$ are equivalent, that is $A=B$; equivalently $A\subset B$ and $B\subset A$.

> Superset. If $A$ is a subset of $B$, then $B$ is a superset of $A$:

$$
A\subset B\Rightarrow B\supset A.
$$

> Disjoint sets. Two sets are disjoint if they have no elements in common, that is $A\cap B=\varnothing$.

> Union. The union of two or more sets is the set containing all possible elements of all the sets, e.g.

$$
A\cup B={a,b,c,e}.
$$

> Intersection. The intersection of two sets $A$ and $B$ is the set of all elements common to $A$ and $B$; written $A\cap B$.

> Difference. The difference $A-B$ is the set of all elements that are in $A$ but not in $B$:

$$
A-B={w:w\in A,\ w\notin B}=A\cap B^{c}.
$$

> Complement. Let $A$ be a subset of the universal set $U$. The complement of $A$, written $A^{c}$, is the set of all elements of $U$ that are not in $A$:

$$
A^{c}={w:w\in U,\ w\notin A}=U-A.
$$

> Power set. Let $A$ be a set. The power set of $A$ is the class consisting of all possible subsets of $A$.

For $A={a,b,c}$ the subsets are organised by the number of elements they contain.

The last row shows that a set is always a subset of itself. Hence

$$
\mathcal{P}(A)={\varnothing,{a},{b},{c},{a,b},{a,c},{b,c},{a,b,c}}.
$$

2.2 Sequences of Sets and Their Limits

A sequence of sets $A_{1},A_{2},\dots,A_{n},\dots$ is denoted by ${A_{n},\ n=1,2,\dots}$ or simply by ${A_{n}}$.

> Monotone increasing. A sequence is monotone increasing if

$$
A_{n}\subseteq A_{n+1}\quad\text{for each }n.
$$

For such a sequence the partial unions satisfy

$$
\bigcup_{i=1}^{n}A_{i}=A_{n},
$$

and the sequence is denoted by $A_{n}\uparrow A$. Its limit is the union of all the terms:

$$
\lim_{n\to\infty}A_{n}=\bigcup_{n=1}^{\infty}A_{n}=A.
$$

> Monotone decreasing. If on the other hand $A_{n}$ is a superset of $A_{n+1}$, that is

$$
A_{n}\supseteq A_{n+1}\quad\text{for every }n,
$$

> then the sequence is monotone decreasing.

For such a sequence

$$
\bigcap_{i=1}^{n}A_{i}=A_{n},
$$

and the limit is the intersection of all the terms:

$$
\lim_{n\to\infty}A_{n}=\bigcap_{n=1}^{\infty}A_{n}=A,\qquad\text{written }A_{n}\downarrow A.
$$

2.3 Fields

> 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 $A\cap B\in\mathcal{F}$.

> 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 $A$.

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

> 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.

> 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.

2.4 DeMorgan's Rule

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

The proof is given in Section 3. The underlying identity used is

$$
(A^{c}\cap B^{c})^{c}=A\cup B.
$$

2.5 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 $X$. The principal types are:

1. one-to-one functions — 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 $w'$.

> Real-valued functions. These are functions which map into the real line, $X:U\to\mathbb{R}$.

> Set functions. These are functions whose domain contains sets, that is, they operate on sets.

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

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

2.6 Probability and Expectation: Notation Fixed at This Stage

The unit fixes the ambient structure $(U,\mathcal{F},P)$: $U$ is the universal (sample) set, $\mathcal{F}$ is the $\sigma$-field of events, and $P$ is the probability measure. The Borel $\sigma$-field $\mathcal{B}$ of the real line is the $\sigma$-field generated by the intervals listed in Section 2.3. The indicator function $I_{A}$ of a set $A$ is the set function that records membership of $A$; it is the elementary building block on which the later treatment of expectation rests. The detailed development of probability measures and of expectation is not part of the present unit, and no additional axioms or formulas are introduced here.