Question

Suppose \(\mathcal{D}\) is a nonempty collection of subsets of \(\mathcal{C}\). Consider the collection of events, $$\mathcal{B}=\cap\\{\mathcal{E}: \mathcal{D} \subset \mathcal{E} \text { and } \mathcal{E} \text { is a } \sigma \text { -field }\\}$$ Note that \(\phi \in \mathcal{B}\) because it is in each \(\sigma\) -field, and, hence, in particular, it is in each \(\sigma\) -field \(\mathcal{E} \supset \mathcal{D}\). Continue in this way to show that \(\mathcal{B}\) is a \(\sigma\) -field.

Short answer

The intersection of all σ-fields containing a certain collection of subsets, denoted \( \mathcal{B} \), is itself a σ-field. This is because it is closed under complements and countable unions, and contains the empty set \( \phi \).

Step-by-step solution

Understand the Definitions

Remember, a collection of subsets \( \mathcal{E} \) is called a σ-field if it fulfills three conditions: (1) The empty set \( \phi \) is an element of \( \mathcal{E} \). (2) If \( A \) is an element of \( \mathcal{E} \) then its complement \( A^c \) is also an element of \( \mathcal{E} \). (3) If \( A_1, A_2, ... \) are elements of \( \mathcal{E} \) then their countable union \( \bigcup_{i=1}^{\infty}A_i \) is also an element of \( \mathcal{E} \).

Proving \( \phi \) Element

Our task is to show that \( \mathcal{B} \) is a σ-field. Note that \( \phi \) is in each σ-field \( \mathcal{E} \), hence, it is in the intersection, \( \mathcal{B} \). This shows that \( \mathcal{B} \) satisfies condition (1) of being a σ-field.

Proving Complement Property

To show closure under complements, assume \( A \) is in \( \mathcal{B} \). Then \( A \) is in every \( \mathcal{E} \). Since \( \mathcal{E} \) is a σ-field, then \( A^c \) is also in every \( \mathcal{E} \). Hence, \( A^c \) is in \( \mathcal{B} \) showing that \( \mathcal{B} \) satisfies condition (2) of being a σ-field.

Proving Countable Unions Property

Lastly, to show closure under countable unions, suppose \( A_1, A_2, ... \) are events in \( \mathcal{B} \). Then, for each \( \mathcal{E} \), since \( \mathcal{E} \) is a σ-field, \( \bigcup_{i=1}^{\infty}A_i \) is in \( \mathcal{E} \). Hence, \( \bigcup_{i=1}^{\infty}A_i \) is in the intersection, \( \mathcal{B} \). This satisfaction of condition (3) completes our proof that \( \mathcal{B} \) is a σ-field.

Key concepts

Set Theory

Set theory is foundational in understanding how we organize and describe collections of objects, broadly referred to as 'sets'. Sets are crucial in various mathematical disciplines, including probability and statistics. A set can be anything you can collect or gather, such as numbers, letters, symbols, or even more sets. In our exercise, the main focus is on a specific type of collection called the \(\sigma\)-field.

A \(\sigma\)-field (or \(\sigma\)-algebra) is a special kind of set collection which follows three essential rules:

• The empty set \(\phi\) is always included in a \(\sigma\)-field.
• Whenever a set \(A\) is part of a \(\sigma\)-field, so must be its complement \(A^c\).
• If you take a countable collection of sets from a \(\sigma\)-field and form a union, this new set is also part of the \(\sigma\)-field.

These conditions ensure the structure and stability of a \(\sigma\)-field, making it a fundamental concept in probability and statistics. The exercise you saw focuses on confirming that a particular collection of events, named \(\mathcal{B}\), is indeed a \(\sigma\)-field by proving it meets all these criteria.

Probability Theory

Probability theory uses the language of set theory to model real-world phenomena where uncertainty or randomness is involved. Within this context, events are sets, and probability is a measure that assigns a likelihood to these events.

A key concept in probability theory is the \(\sigma\)-field, as it forms the underlying structure in which probabilities are defined. A \(\sigma\)-field of events allows us to ensure that probabilities are consistently assigned to sets belonging to it. This consistency is crucial for calculating probabilities accurately.

• The core idea is to have a collection of sets, such as the \(\sigma\)-field, which is stable under the operations (like taking complements and countable unions) commonly used when calculating probabilities.
• For any event \(A\) in a \(\sigma\)-field, we can also be sure any countable combination of its sub-events behaves predictably and has a well-defined probability.
• Our exercise examined a specific collection \(\mathcal{B}\), confirming it is a \(\sigma\)-field, thus suitable for defining a valid probability distribution over it.

Understanding probability theory involves making such logical connections and structures intuitive and applying them to solve problems effectively.

Mathematical Statistics

Mathematical statistics builds on the foundations of probability theory to make informed inferences about populations from sample data. As we already know from probability theory, properly understanding \(\sigma\)-fields is necessary because they ensure the events we consider are manageable and coherent with probabilistic laws.

Here's how this ties into statistics:

• Inference in statistics often involves considering complex events, such as finding the likelihood of a particular sample outcome. Having a \(\sigma\)-field means we can confidently assign probabilities to these complex, compound events.
• Statisticians use \(\sigma\)-fields to ensure that the samples and data they test can be treated in calculations that yield valid confidence intervals and hypothesis tests.
• The exercise illustrates verifying collections, like \(\mathcal{B}\), as \(\sigma\)-fields. This verification is crucial for using these collections in statistical models where consistent and reliable analyses are required.

With mathematical statistics, \(\sigma\)-fields help to bridge the gap between theoretical models and practical data analysis, reinforcing statistical methods with mathematical rigor.