2.1 Measure Theory

This deals with the issue of assigning a measure (e.g. length, area volume) to certain sets. Properties we should expect form a measure $\mu$ are:

2.1.1 Properties

• Well-definedness: it should take values in $[0, \infty]$ and $\mu(\emptyset) = 0$.
• Additiveness: if $A \cup B = \emptyset$ then $\mu(A \cup B) = \mu(A) + \mu(B)$.
• Invariant (addtional): under congruences and translations as Lebesgue measure on $R^n$.

Imagine that using these properties we try to obtain the area of a circle. We will notice that finite additivity is not enough, we also need to introduce $sigma$-additivity on additiveness property.

2.1.2 Set algebra and countability

$$\begin{align} A \cup B &= \{x: x \in A ~\text{or}~ x \in B ~\text{or}~ x \in A ~\text{and}~ B\}\\ A \cap B &= \{x: x \in A ~\text{and}~ B\}\\ A \setminus B &= \{x: x \in A ~\text{and}~ B\} \end{align}$$

• $A \dot{\cup} B$ represents disjoint union,
• $A \subset B$ means A is contained in B including $A = B$.
• $A^c := X \setminus A$ for $A \subset B$ is the complement of A relative to $X$.

Distributive laws: $$\begin{align} A \cap (B \cup C) & = (A \cap B) \cup (A \cap C) \\ A \cap (B \cup C) & = (A \cap B) \cup (A \cap C) \end{align}$$

Morgan’s identities $$\begin{align} \left(\bigcap_{i \in I} A_i\right)^c &= \bigcup_{i \in I} A_i^c \\ \left(\bigcup_{i \in I} A_i\right)^c &= \bigcap_{i \in I} A_i^c \end{align}$$

A function $f: X \rightarrow Y$ is:

• injective (one-to-one) $\Leftrightarrow$ $f(x) = f(x')$ implies $x = x'$,
• surjective (onto) $\Leftrightarrow$ $f(X) := {f(x) \in Y: x \in X} = Y$,
• bijective $\Leftrightarrow$ $f(.)$ is injective and surjective.

Set operations shown before and direct images under a funtion $f$ are not necessarily compatible: $$\begin{align} f(A \cup B) &= f(A) \cup f(B) \\ f(A \cap B) &\neq f(A) \cap f(B) \\ f(A \setminus B) &\neq f(A) \setminus f(B) \\ \end{align}$$ While inverse images and set operations are always compatible. The inverse mapping $f_{-1}$ maps subsets $C \subset Y$ into subsets of $X$: $$\begin{equation} f^{-1}(C) := \{x \in X: f(x) \in C\} \in X, ~\text{for all}~ C \subset Y. \end{equation}$$ It follows that: $$\begin{align} f^{-1}(\bigcup_{i \in I} C_i) &= \bigcup_{i \in I} f^{-1}(C_i), \\ f^{-1}(\bigcap_{i \in I} C_i) &= \bigcap_{i \in I} f^{-1}(C_i), \\ f^{-1}(C \setminus D) &= f^{-1}(C) \setminus f^{-1}(D) \end{align}$$

2.1.3 $\sigma$-Algebras

A measure function must be defined on a system of sets stable under repetition of set operations ($\cup, \cap, ^c$) countably many times.

A $\sigma$-algebra $\mathcal{A}$ on a set $X$ is a family of subsets of $X$ with the following properties: $$\begin{align} X &\in \mathcal{A}, \\ A \in \mathcal{A} & \Rightarrow A^c \in \mathcal{A}, \\ (A_n)_{n\in N} \subset \mathcal{A} & \Rightarrow \bigcup_{n \in N} A_n \in \mathcal{A}. \end{align}$$

Based on these definitions, we can obtain some properties:

• $\emptyset \in \mathcal{A}$
• $A, B \in \mathcal{A} \Rightarrow A \cup B \in \mathcal{A}$
• $(A_n)_{n \in N} \subset \mathcal{A} \Rightarrow \cap_{n \in N} A_n \in \mathcal{A}$