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Chapter 10: Problem 2

Let \(\mathbf{R}^{2}=\mathbf{R} \times \mathbf{R}\), and let \(\mathcal{B}^{2}\) be the Borel sets of \(\mathbf{R}^{2}\), while \(\mathcal{B}\) denotes the Borel sets of \(\mathbf{R}\). Show that \(\mathcal{B}^{2}=\mathcal{B} \otimes \mathcal{B}\).

Short Answer

Expert verified
The Borel \( \sigma \)-algebra on \( \mathbb{R}^2 \) is equivalent to the product \( \sigma \)-algebra of Borel sets on \( \mathbb{R} \).

Step by step solution

01

Understand the Borel Sets

Borel sets, denoted by \( \mathcal{B} \), are the \( \sigma \)-algebra generated by open sets in \( \mathbb{R} \). \( \mathcal{B}^{2} \), therefore, represents the \( \sigma \)-algebra generated by open rectangles in \( \mathbb{R}^2 \).
02

Introduction to the Product \( \sigma \)-algebra

The product \( \sigma \)-algebra \( \mathcal{B} \otimes \mathcal{B} \) is defined as the smallest \( \sigma \)-algebra containing all sets of the form \( A \times B \), where \( A, B \in \mathcal{B} \).
03

Show Inclusion \( \mathcal{B}^{2} \subseteq \mathcal{B} \otimes \mathcal{B} \)

Since any open rectangle in \( \mathbb{R}^2 \) can be written as a countable union and intersection of sets of the form \( A \times B \), where each \( A \) and \( B \) are open sets in \( \mathbb{R} \), and such sets \( A \times B \) are in \( \mathcal{B} \otimes \mathcal{B} \), it follows that \( \mathcal{B}^{2} \), which is generated by these open rectangles, must be a subset of \( \mathcal{B} \otimes \mathcal{B} \).
04

Show Inclusion \( \mathcal{B} \otimes \mathcal{B} \subseteq \mathcal{B}^{2} \)

To prove the opposite inclusion, consider that \( \mathcal{B} \otimes \mathcal{B} \) by definition contains \( A \times B \) for \( A, B \in \mathcal{B} \). These are precisely the building blocks of \( \mathcal{B}^{2} \), as any set in \( \mathcal{B} \) is also a Borel set in \( \mathbb{R} \), and therefore their products lie in \( \mathcal{B}^{2} \). As \( \mathcal{B}^{2} \) is closed under countable unions and intersections, it must contain \( \mathcal{B} \otimes \mathcal{B} \).
05

Conclude the Equality

Since we've shown \( \mathcal{B}^{2} \subseteq \mathcal{B} \otimes \mathcal{B} \) and \( \mathcal{B} \otimes \mathcal{B} \subseteq \mathcal{B}^{2} \), the two \( \sigma \)-algebras must be equal. Therefore, \( \mathcal{B}^{2} = \mathcal{B} \otimes \mathcal{B} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sigma-Algebra
A sigma-algebra, sometimes denoted as a \( \sigma \)-algebra, is a mathematical structure used in measure theory, a branch of mathematics that investigates sizes or measures of sets. To get a good grasp of \( \sigma \)-algebras, think of them as a collection of sets that satisfy certain properties.Here are the main features of a \( \sigma \)-algebra:
  • It includes the universal set and the empty set. In our case of Borel sets, these are all possible sets in \( \mathbb{R} \) or \( \mathbb{R}^2 \), and the set containing no elements.
  • Closed under complementation: If a set is in the \( \sigma \)-algebra, its complement must also be in the \( \sigma \)-algebra. This means if you have a set of numbers, the set of numbers not in it is also included.
  • Closed under countable unions: If you have a countable collection of sets in the \( \sigma \)-algebra, their union will also be a set in the \( \sigma \)-algebra.
These properties make \( \sigma \)-algebra essential in defining measures, like probability measures, since they provide the framework necessary to talk about the size of sets consistently.
Product Sigma-Algebra
The product \( \sigma \)-algebra is a special type of \( \sigma \)-algebra formed from two \( \sigma \)-algebras. In our example, the product \( \sigma \)-algebra \( \mathcal{B} \otimes \mathcal{B} \) is formed from the Borel sets of \( \mathbb{R} \), denoted as \( \mathcal{B} \).Here is how a product \( \sigma \)-algebra works:
  • It includes all sets formed by taking the Cartesian product of two Borel sets, one from each \( \sigma \)-algebra. For instance, if you have a set \( A \) from \( \mathcal{B} \) and a set \( B \) from another \( \mathcal{B} \), \( A \times B \) is in the product \( \sigma \)-algebra \( \mathcal{B} \otimes \mathcal{B} \).
  • This product includes enough sets to satisfy \( \sigma \)-algebra properties on a multi-dimensional level, like being closed under countable unions, intersections, and complements.
  • It reflects the idea that you are combining two spaces, \( \mathbb{R} \) and another \( \mathbb{R} \), resulting in a space like \( \mathbb{R}^2 \).
The notion of a product \( \sigma \)-algebra becomes particularly significant in the analysis of multidimensional spaces, as it allows for the definition of joint measures across different dimensions.
Open Sets
Open sets are fundamental in the study of topology and serve as the building blocks for constructing Borel sets, denoted as \( \mathcal{B} \). Simply put, these are sets where, at every point within the set, you can move around a little bit without "leaving" the set.Why are open sets important?
  • Open sets allow for the construction of the Borel \( \sigma \)-algebra, which includes all sets that can be formed through countable operations (unions, intersections, and complements) from open sets.
  • In \( \mathbb{R}^2 \), the idea of open sets extends to open rectangles. These are regions where, similar to open intervals, every point has a neighborhood entirely contained within the rectangle.
  • Understanding open sets is crucial for grasping more advanced concepts in metric spaces and analysis. In essence, they provide a way to discuss continuity, limits, and convergence.
Recognizing open sets and how they relate to \( \sigma \)-algebras helps unravel more complex structures within mathematics, like Borel sets and their role in measure theory.

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Most popular questions from this chapter

Let \(X, Y\) be independent random variables taking values in \(\mathbf{N}\) with $$ P(X=i)=P(Y=i)=\frac{1}{2^{i}} \quad(i=1,2, \ldots) $$ Find the following probabilities: a) \(P(\min (X, Y) \leq i) \quad\left[\right.\) Ans.: \(\left.1-\frac{1}{4^{4}}\right]\) b) \(P(X=Y)\) [Ans.: \(\frac{1}{3}\) ] c) \(P(Y>X) \quad\left[\right.\) Ans.: \(\left.\sum_{i \geq 0}^{3} \frac{1}{2^{2}\left(2^{i}-1\right)}\right]\) d) \(P(X\) divides \(Y)\) [Ans,: \(\frac{1}{3}\) ] e) \(P(X \geq k Y)\) for a given positive integer \(k\) [Ans,: \(\frac{1}{2^{1+k}-1}\) ]
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