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1 — Cardinality, Cantor’s Theorem, Series, Measure Theory

Lecture 01 covers early measure-theoretic foundations: countable vs uncountable sets, Cantor’s diagonal argument (including $vert A\vert <\vert 2^A\vert $), binary expansions showing ($(0,1)$) is uncountable, basics of sequences/series (convergence, harmonic vs (p)-series, alternating divergence), and first steps in measure theory—definition of an algebra of sets and a measure on it.

1. Cardinality of Sets

A set is finite, countably infinite, or uncountable.

Definition (Cardinality)

For a set $A$, the cardinality $\vert A\vert $ is the “number of elements” in $A$.

  • If there is a bijection $A \leftrightarrow \mathbb{N}$, then $A$ is countable.
  • If no such bijection exists, $A$ is uncountable.

Examples of Countable Sets

  • $\mathbb{N} = {1,2,3,\dots}$
  • $\mathbb{Z} = {\dots,-2,-1,0,1,2,\dots}$
  • $\mathbb{Q} = {\frac{m}{n}: m,n\in\mathbb{Z}, n\neq 0}$

A typical enumeration of $\mathbb{Q}$ uses the lattice of integer pairs $(m,n)$ arranged in diagonals.

Real numbers are uncountable

\(\vert \mathbb{R}\vert > \vert \mathbb{Z}\vert .\)

This is shown using Cantor’s diagonal argument.

2. Cantor’s Theorem: $\vert A\vert < \vert 2^A\vert $

Power Set

For any set $A$, the power set \(2^A = \{B : B\subseteq A\}\) has cardinality strictly greater than $A$.

Theorem (Cantor)

For every set $A$, \(\vert A\vert < \vert 2^A\vert .\)

Proof (Diagonal Argument)

Assume for contradiction that there is a bijection
\(f : A \to 2^A.\)

Define the set \(B = \{a\in A : a\notin f(a)\}.\)

Since $B\subseteq A$, if $f$ is onto, there must be some $b\in A$ such that \(f(b) = B.\)

Now consider whether $b\in B$:

  • If $b\in B$, then by definition of $B$, $b\notin f(b)=B$. Contradiction.
  • If $b\notin B$, then by definition of $B$, $b\in f(b)=B$. Contradiction.

Thus no bijection exists and $\vert A\vert < \vert 2^A\vert $.

3. Binary Expansions and the Uncountability of $(0,1)$

Every number in $(0,1)$ can be written as a binary expansion: \(x = \sum_{k=1}^\infty \frac{\varepsilon_k}{2^k}, \qquad \varepsilon_k \in \{0,1\}.\)

This identifies $(0,1)$ with a subset of ${0,1}^{\mathbb{N}}$.
Because ${0,1}^{\mathbb{N}}$ is uncountable, $(0,1)$ is uncountable.

4. Sequences and Series

Sequence

A sequence is a function \(a : \mathbb{N} \to \mathbb{R}, \qquad n\mapsto a_n.\)

Partial Sums and Series

The partial sums of a series are \(S_N = \sum_{i=1}^N a_i.\)

The series $\sum_{n=1}^\infty a_n$ converges if $\lim_{N\to\infty} S_N$ exists and is finite.

Basic Facts

  • If $a_n \ge 0$ and $\sum a_n < \infty$, then $S_n$ is increasing and bounded, so it converges.
  • Example:
    • Harmonic series diverges: \(\sum_{n=1}^{\infty} \frac{1}{n} = \infty.\)
    • $p$-series with $p>1$ converges: \(\sum_{n=1}^{\infty} \frac{1}{n^2} < \infty.\)
  • Alternating series like $(-1)^n$ do not converge, because partial sums oscillate.

5. Measure Theory: First Definitions

Let $\Omega$ be a set.

Algebra of Sets

A collection $\mathcal{A}\subseteq 2^\Omega$ is an algebra if:

  1. $A,B \in \mathcal{A} \implies A\cap B \in \mathcal{A}$.
  2. $A\in \mathcal{A} \implies A^c \in \mathcal{A}$.
  3. $\varnothing \in \mathcal{A}$, hence $\Omega\in\mathcal{A}$.
  4. Closure under union follows from De Morgan: \(A\cup B = (A^c \cap B^c)^c.\)

Example

On $\Omega = \mathbb{R}$, let $\mathcal{A}$ be the set of finite unions of half-open intervals: \(\mathcal{A} = \bigcup_{k=1}^N (a_k,b_k], \quad \text{disjoint intervals}.\)

This is an algebra.

6. Measures

A measure on an algebra $\mathcal{A}$ is a function \(\mu : \mathcal{A} \to [0,\infty]\) such that:

  1. $\mu(A)\ge 0$ for all $A\in\mathcal{A}$.
  2. $\mu(\varnothing)=0$.
  3. Countable additivity for disjoint sets:
    If $A_i \in \mathcal{A}$ are pairwise disjoint and $\bigcup_{i=1}^\infty A_i \in \mathcal{A}$, then \(\mu\!(\bigcup_{i=1}^\infty A_i) = \sum_{i=1}^\infty \mu(A_i).\)

This is the foundation for the probability measure $P$ later in the course.

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