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