Suppose A, B are independent events (in a probability space). Show that any two events C ∈ σ(A) and D ∈ σ(B) are independent.

Given that \( A \) and \( B \) are independent events, we want to show that any two events \( C \in \sigma(A) \) and \( D \in \sigma(B) \) are independent, where \( \sigma(A) \) and \( \sigma(B) \) denote the sigma-algebras generated by \( A \) and \( B \), respectively.

### Step 1: Understanding \( \sigma(A) \) and \( \sigma(B) \)
- The sigma-algebra \( \sigma(A) \) consists of all events that can be expressed in terms of \( A \), meaning it contains sets like \( A \), \( A^c \), \( \emptyset \), and the entire sample space \( \Omega \).
- Similarly, \( \sigma(B) \) contains all events that can be expressed in terms of \( B \), such as \( B \), \( B^c \), \( \emptyset \), and \( \Omega \).

### Step 2: Definition of Independence
Two events \( C \in \sigma(A) \) and \( D \in \sigma(B) \) are independent if:
\[
P(C \cap D) = P(C) \cdot P(D).
\]
We are tasked with proving that this holds for any \( C \in \sigma(A) \) and \( D \in \sigma(B) \).

### Step 3: Independence of \( A \) and \( B \)
By assumption, \( A \) and \( B \) are independent, which means:
\[
P(A \cap B) = P(A) \cdot P(B),
\]
and similarly for other combinations like \( A \cap B^c \), \( A^c \cap B \), and \( A^c \cap B^c \).

### Step 4: Using a Generating Class Argument
The sigma-algebra \( \sigma(A) \) is generated by the set \( \{A, A^c\} \), and \( \sigma(B) \) is generated by \( \{B, B^c\} \). Thus, every event in \( \sigma(A) \) can be written as a finite union of intersections of the sets \( A \) and \( A^c \), and similarly for \( \sigma(B) \).

To prove the independence of arbitrary events in \( \sigma(A) \) and \( \sigma(B) \), we can use the fact that the independence of generating events (such as \( A \) and \( B \), \( A \) and \( B^c \), etc.) implies the independence of all events in the sigma-algebras generated by them.

### Step 5: Verifying Independence for Generating Events
For any \( C \in \sigma(A) \) and \( D \in \sigma(B) \), we know that \( C \) is a union of intersections of the events \( A \) and \( A^c \), and \( D \) is a union of intersections of \( B \) and \( B^c \). Since the independence property holds for \( A \) and \( B \), it extends to any finite union or intersection of these events. Therefore, \( C \) and \( D \) must also be independent.

### Conclusion:
We have shown that for any events \( C \in \sigma(A) \) and \( D \in \sigma(B) \), the probability of their intersection is the product of their probabilities:
\[
P(C \cap D) = P(C) \cdot P(D),
\]
proving that \( C \) and \( D \) are independent.

---

Would you like further clarification on any step? Here are five related questions for deeper understanding:

1. What are some examples of events in \( \sigma(A) \) and \( \sigma(B) \)?
2. How can the concept of generating sets be applied to other fields of mathematics?
3. Can you explain the difference between pairwise and mutual independence for more than two events?
4. How does the assumption of independence affect the structure of the probability space?
5. What are some practical applications of sigma-algebras in probability theory?

**Tip:** Independence in probability theory is a powerful concept, but always verify the conditions carefully, as subtle differences in assumptions can lead to significant changes in outcomes.

Learn how to prove the independence of events C ∈ σ(A) and D ∈ σ(B), where A and B are independent events in a probability space. This solution involves the concepts of sigma-algebras, generating classes, and independence, and uses key probability formulas. Suitable for advanced students in probability theory.

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