This section introduces the essential elements of set theory, which are crucial for understanding Brownian motion, random variables, probability theory, information economics, and game theory. Since set theory is not sufficiently covered in formal economic training, we provide a detailed overview here. Set theory involves defining sets, which are collections of objects. Sets are denoted by curly brackets, with elements listed inside. For example, the set of real numbers x satisfying ax + bx² ≥ 0 is written as M := {x ∈ ℝ | ax + bx² ≥ 0}. A set with a single element is called a point set. For real numbers, abbreviations are often used, such as (0,1) for the interval of numbers between 0 and 1. The set of real numbers is denoted by ℝ, natural numbers by ℕ, integers by ℤ, and rational numbers by ℚ. The empty set is denoted by ∅. Sets are unordered, but the order of elements can be important in some contexts, such as tuples and pairs. Set operations include union (denoted by ∪), intersection (denoted by ∩), and difference (denoted by \). For example, the union of {1,2} and {3,4} is {1,2,3,4}, while the intersection of {1,2} and {2,3} is {2}. Disjoint sets have no elements in common. A subset B of a set A contains only elements from A. The difference A \ B contains elements of A not in B. Set operations follow rules similar to arithmetic, such as A \ (B ∩ C) = (A \ B) ∪ (A \ C). These rules can be visualized using Venn diagrams. Set operations also include infinite unions and intersections. For example, the infinite union of sets A₁, A₂, ... is the set containing all elements from each set. The infinite intersection is the set containing elements common to all sets. The power set of a set Ω is the set of all subsets of Ω, denoted by P(Ω). It is much larger than Ω itself. For example, the power set of {1,2,3,4,5,6} has 64 subsets. In economics, events are often described using sets. An event is a subset of the event space Ω. For example, the event of rolling a 1 on a die is {1}. Events can be elementary (containing a single outcome) or composite (containing multiple outcomes). In discrete-time models, events are described by sequences of outcomes, such as (u, d) for a coin toss. In continuous-time models, events are described by continuous functions, such as the evolution of a share price over time. Brownian motion uses the event space C[0, ∞), which consists of all continuous functions fromThis section introduces the essential elements of set theory, which are crucial for understanding Brownian motion, random variables, probability theory, information economics, and game theory. Since set theory is not sufficiently covered in formal economic training, we provide a detailed overview here. Set theory involves defining sets, which are collections of objects. Sets are denoted by curly brackets, with elements listed inside. For example, the set of real numbers x satisfying ax + bx² ≥ 0 is written as M := {x ∈ ℝ | ax + bx² ≥ 0}. A set with a single element is called a point set. For real numbers, abbreviations are often used, such as (0,1) for the interval of numbers between 0 and 1. The set of real numbers is denoted by ℝ, natural numbers by ℕ, integers by ℤ, and rational numbers by ℚ. The empty set is denoted by ∅. Sets are unordered, but the order of elements can be important in some contexts, such as tuples and pairs. Set operations include union (denoted by ∪), intersection (denoted by ∩), and difference (denoted by \). For example, the union of {1,2} and {3,4} is {1,2,3,4}, while the intersection of {1,2} and {2,3} is {2}. Disjoint sets have no elements in common. A subset B of a set A contains only elements from A. The difference A \ B contains elements of A not in B. Set operations follow rules similar to arithmetic, such as A \ (B ∩ C) = (A \ B) ∪ (A \ C). These rules can be visualized using Venn diagrams. Set operations also include infinite unions and intersections. For example, the infinite union of sets A₁, A₂, ... is the set containing all elements from each set. The infinite intersection is the set containing elements common to all sets. The power set of a set Ω is the set of all subsets of Ω, denoted by P(Ω). It is much larger than Ω itself. For example, the power set of {1,2,3,4,5,6} has 64 subsets. In economics, events are often described using sets. An event is a subset of the event space Ω. For example, the event of rolling a 1 on a die is {1}. Events can be elementary (containing a single outcome) or composite (containing multiple outcomes). In discrete-time models, events are described by sequences of outcomes, such as (u, d) for a coin toss. In continuous-time models, events are described by continuous functions, such as the evolution of a share price over time. Brownian motion uses the event space C[0, ∞), which consists of all continuous functions from