Sequences And Limits

1. Core Concept Overview & Intuition

The study of sequences and limits begins with the elementary algebra of sets and builds toward the notion of a limit of a sequence of sets. The central intuition is that a sequence of sets can grow toward a limiting set (monotone increasing) or shrink toward a limiting set (monotone decreasing). In the increasing case, each set is contained in the next, so the partial unions stabilize at the current term; in the decreasing case, each set contains the next, so the partial intersections stabilize at the current term. The limit is then the union of all terms (increasing case) or the intersection of all terms (decreasing case).

This limiting behaviour is the foundation for the structural notions that follow. A field is a class of subsets closed under complementation and finite intersections; DeMorgan's rule extends this closure to finite unions. Adding closure under countable operations produces a $\sigma$-field, and the equivalence between $\sigma$-fields and monotone fields is the key bridge between the sequential limit idea and the algebraic closure idea. Functions, particularly inverse images and indicator functions, then provide the language for describing sets and their combinations, and simple functions are shown to be expressible as finite sums over pairwise disjoint sets.

Throughout, the guiding principle is that limits of monotone sequences of sets are computed by taking the union (increasing) or intersection (decreasing) of all terms, and that closure properties are verified by reducing a new operation to the two primitive operations of complementation and intersection.

2. Mathematical Formalization & Core Definitions

2.1 Set Operations

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

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

> Definition (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 $A$:

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

> Definition (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 organized by the number of elements they contain:

The last entry 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

> Definition (Sequence of Sets). A sequence of sets $A_{1},A_{2},\dots,A_{n},\dots$ is denoted by ${A_{n},\ n=1,2,\dots}$ or simply by ${A_{n}}$.

> Definition (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:

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

> Definition (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:

$$
\lim_{n\to\infty}A_{n}=\bigcap_{n=1}^{\infty}A_{n}=A,
$$

> written $A_{n}\downarrow A$.

The essential computational content is that for a monotone sequence the limit is obtained by taking the union of all terms (increasing) or the intersection of all terms (decreasing). The partial unions and partial intersections stabilize at the current term, which is what makes the limit computable term by term.

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

> 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:

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

> 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.

2.4 Functions

> Definition (Function). 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

$$
X:U\to U'.
$$

> $U$ is the domain and $U'$ the range (codomain) of $X$.

The source distinguishes three elementary types of mapping:

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 $w'$.

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

> Definition (Set Function). These are functions whose domain contains sets, that is, they operate 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.

> 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 $U$.

2.5 Probability

> Definition (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

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

The source also records the elementary vocabulary of an experiment:

> Definition (Experiment, Sample Space, Outcome). An experiment is an activity whose outcome cannot be determined with certainty. The sample space is the set of all possible outcomes. An outcome is an element of the sample space.

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

$$
S={H,T},\quad E_{1}={\varnothing},\quad E_{2}={H},\quad E_{3}={T},\quad E_{4}={H,T}.
$$