2  \(\sigma\)-Field in Probibility

2.1 \(\sigma\)-Field (\(\sigma\)-Algebra)

A \(\sigma\)-field (or \(\sigma\)-algebra) is a collection of subsets of a given set \(X\) that satisfies certain properties. In the context of probability theory, \(\sigma\)-fields are crucial because they define which subsets of the sample space have measurable probabilities.

A collection \(\mathcal{F}\) of subsets of \(X\) is a \(\sigma\)-field if it satisfies the following three properties:

  1. Closure under Complement: If \(A\) is in \(\mathcal{F}\), then its complement, \(X \setminus A\), is also in \(\mathcal{F}\). In other words, if \(A \in \mathcal{F}\), then \(\overline{A} = X \setminus A \in \mathcal{F}\).

  2. Closure under Countable Unions: If \(A_1, A_2, A_3, \ldots\) (a countable sequence) are in \(\mathcal{F}\), then their union \(\bigcup_{i=1}^{\infty} A_i\) is also in \(\mathcal{F}\). This property ensures that \(\sigma\)-fields are closed under countable unions of sets.

  3. Closure under Countable Intersections (Optional Property): If \(A_1, A_2, A_3, \ldots\) (a countable sequence) are in \(\mathcal{F}\), then their intersection \(\bigcap_{i=1}^{\infty} A_i\) is also in \(\mathcal{F}\). This property is sometimes included in the definition of a \(\sigma\)-field, but it can be derived from properties 1 and 2.

\(\sigma\)-fields are used in probability theory to define measurable spaces. In the context of probability, the elements of the \(\sigma\)-field represent events, and any subset of the sample space that belongs to the \(\sigma\)-field can have a probability assigned to it. \(\sigma\)-fields provide a formal and rigorous way to define the structure of events to which probabilities can be assigned in a consistent manner.

2.2 Examples of \(\sigma\)-Field in Probability Theory

2.2.1 Trivial \(\sigma\)-Field

In any given probability space, the smallest \(\sigma\)field, denoted as \(\{\emptyset, S\}\), contains only the empty set (\(\emptyset\)) and the entire sample space (\(S\)). This \(\sigma\)field, called the trivial \(\sigma\)-field, is the smallest possible \(\sigma\)field because it has only two elements.

2.2.2 Power Set

The largest \(\sigma\)field, also known as the power set of the sample space, denoted as \(\mathcal{P}(S)\), contains all possible subsets of the sample space \(S\), including the empty set and \(S\). The power set includes every possible combination of outcomes and is the largest \(\sigma\)field because it contains all possible events that can be defined based on the sample space.

2.2.3 Tossing a Biased Coin

Imagine tossing a biased coin where the probability of landing heads (\(H\)) is \(0.6\) and the probability of landing tails (\(T\)) is \(0.4\).

The sample space \(S\) for this experiment consists of two outcomes: \(S = \{H, T\}\).

A \(\sigma\)-field \(\mathcal{F}\) for this experiment could be defined as follows:

  • Individual Outcomes: Both \(H\) and \(T\) are in \(\mathcal{F}\) because they are subsets of the sample space.

  • Complements: If \(H\) is in \(\mathcal{F}\), then its complement \(\overline{H} = \{T\}\) is also in \(\mathcal{F}\). Similarly, if \(T\) is in \(\mathcal{F}\), then its complement \(\overline{T} = \{H\}\) is in \(\mathcal{F}\).

  • Countable Unions: The sets \(\{H\}\) and \(\{T\}\) are in \(\mathcal{F}\), so their union \(\{H\} \cup \{T\} = \{H, T\}\) (the entire sample space) is also in \(\mathcal{F}\).

  • Countable Intersections: The sets \(\{H\}\) and \(\{T\}\) are in \(\mathcal{F}\), so their intersection \(\{H\} \cap \{T\} = \{\}\) (the empty set) is also in \(\mathcal{F}\).

In this example, \(\mathcal{F}\) comprises individual outcomes, their complements, and combinations of outcomes through unions and intersections. Any subset of the sample space and its complements, unions, and intersections that are in \(\mathcal{F}\) can have a probability assigned to them, allowing for the calculation of probabilities associated with various

2.2.4 Rolling a Fair Six-Sided Die

Let’s consider a simple experiment of rolling a fair six-sided die. The sample space, denoted as \(S\), consists of the possible outcomes: \(\{1, 2, 3, 4, 5, 6\}\).

A \(\sigma\)-field \(\mathcal{F}\) for this experiment could be defined as follows:

  • \(\mathcal{F}\) includes all possible subsets of the sample space \(S\). This means \(\mathcal{F}\) contains sets such as \(\{1, 2, 3\}\), \(\{4\}\), \(\{2, 5, 6\}\), and even the empty set \(\{\}\).

  • \(\mathcal{F}\) includes the complements of its subsets. For example, if \(\{1, 2, 3\}\) is in \(\mathcal{F}\), then its complement \(\{4, 5, 6\}\) must also be in \(\mathcal{F}\).

  • \(\mathcal{F}\) includes countable unions and intersections of its subsets. For instance, if both \(\{1, 2\}\) and \(\{2, 3, 4\}\) are in \(\mathcal{F}\), then their union \(\{1, 2\} \cup \{2, 3, 4\} = \{1, 2, 3, 4\}\) and intersection \(\{1, 2\} \cap \{2, 3, 4\} = \{2\}\) must also be in \(\mathcal{F}\).

In this example, \(\mathcal{F}\) represents all the possible events and combinations of events that we can define based on the outcomes of rolling the die. Any subset of the sample space and its complements, unions, and intersections that are in \(\mathcal{F}\) can have a probability assigned to them. This structure allows us to work with probabilities in a well-defined and consistent manner, making \(\sigma\)-fields a fundamental concept in probability theory.