Question

Prove that if each of the measures \(\mu_{1}\) and \(\mu_{2}\) has a countable base, then so does their direct product \(\mu=\mu_{1} \times \mu_{2}\). Comment. In particular, Lebesgue measure in the plane (or more generally in n-space) has a countable base.

Short answer

The product measure \( \mu = \mu_1 \times \mu_2 \) has a countable base because it can be covered by countable products of bases of \( \mu_1 \) and \( \mu_2 \).

Step-by-step solution

Understanding Measure and Countable Bases

To prove that the direct product measure \( \mu = \mu_{1} \times \mu_{2} \) has a countable base, we first need to understand what it means for a measure to have a countable base. A measure \( \mu \) on a set \( X \) has a countable base if there exists a countable collection of sets \( \{E_n\}_{n=1}^\infty \) such that for any measurable set \( A \) and any \( \epsilon > 0 \), there exists a finite union of these sets, \( E = \bigcup_{i=1}^k E_{n_i} \), with \( \mu(A \triangle E) < \epsilon \).

Defining the Countable Base for \( \mu_1 \) and \( \mu_2 \)

Since \( \mu_1 \) and \( \mu_2 \) each have countable bases, there exist countable collections of sets \( \{U_n\}_{n=1}^\infty \) and \( \{V_m\}_{m=1}^\infty \) such that for any measurable set and any \( \epsilon > 0 \), we can find finite unions of these sets that approximate the measurable set with respect to \( \mu_1 \) and \( \mu_2 \), respectively.

Constructing the Countable Base for \( \mu \)

To construct a countable base for the product measure \( \mu = \mu_1 \times \mu_2 \), we consider the collection of sets of the form \( U_n \times V_m \) for all \( n, m \in \mathbb{N} \). This collection is countable because it is the Cartesian product of two countable sets, \( \{U_n\}_{n=1}^\infty \) and \( \{V_m\}_{m=1}^\infty \).

Proving the Countable Base Property for \( \mu \)

For any measurable set \( A \subset X \times Y \) and any \( \epsilon > 0 \), we can approximate \( A \) by a finite union of sets from our constructed base \( \{U_n \times V_m\} \). Given \( A = A_x \times A_y \), find approximations \( U = \bigcup_{i=1}^k U_{n_i} \) for \( A_x \) and \( V = \bigcup_{j=1}^l V_{m_j} \) for \( A_y \) with the measure and error constraints for \( \mu_1 \) and \( \mu_2 \), then \( U \times V \) approximates \( A \) for \( \mu \). The measure \( \mu(A \triangle (U \times V)) < \epsilon \).

Conclusion: Existence of Countable Base for \( \mu \)

Since we've shown the existence of the countable collection \( \{U_n \times V_m\} \) that satisfies the conditions for a countable base for the product measure \( \mu = \mu_1 \times \mu_2 \), we conclude that the direct product of two measures with countable bases also has a countable base.

Key concepts

Countable Base

A countable base for a measure is an essential concept in measure theory, which greatly helps in simplifying complex analysis. It refers to a set of collection that is countable and can adequately represent or approximate any measurable set within a given error margin, which is denoted by \( \epsilon \).
A measure \( \mu \) on a set \( X \) is said to have a countable base if there exists a sequence \( \{E_n\}_{n=1}^{\infty} \) of measurable sets such that for any given measurable set \( A \) and any positive \( \epsilon \), we can find a finite union of these sets that comes within \( \epsilon \) of \( A \) in measure. This implies that for measurement purposes, only countable selections of sets are needed to form approximations.

• This collection serves as a building block for analyses and calculations involving measurable sets.
• The accuracy of this approximation is governed strictly by the chosen error tolerance \( \epsilon \).

The main advantage of having a countable base is that it reduces the complexity inherent in dealing with infinite collections, providing structure and simplicity to the measure processes utilized in further mathematical operations or proofs.

Direct Product Measure

The concept of a direct product measure is instrumental in extending the idea of measure from one dimension to multidimensional spaces. This type of measure combines two separate measures on different sets, resulting in a unified measure that can be applied to the Cartesian product of these sets.
Given two measures, \( \mu_1 \) and \( \mu_2 \), defined on sets \( X \) and \( Y \) respectively, their direct product measure, denoted as \( \mu = \mu_1 \times \mu_2 \), works on the product space \( X \times Y \).

• The collection of sets for this measure is typically constructed from products \( U_n \times V_m \), where \( U_n \) and \( V_m \) come from the countable bases of \( \mu_1 \) and \( \mu_2 \).
• This approach allows the definition of measures over higher-dimensional sets, thereby enabling more comprehensive analyses in spaces such as the plane or n-space.

The direct product measure retains key properties like having a countable base if each original measure does too, thanks to the Cartesian product of the respective countable collections. Hence, combining two measures with countable bases allows the resulting product measure to also possess a countable base.

Lebesgue Measure

Lebesgue measure is one of the most fundamental concepts in real analysis and measure theory. It extends the notion of length in the 1-dimensional space to more complex n-dimensional spaces. Its utility lies in its ability to measure the "volume" or "size" of potentially complicated sets, even in infinite-dimensional spaces.
The primary characteristic of the Lebesgue measure is its compatibility with limiting processes, making it vital for integration in modern analysis. It helps in defining integrals of functions over complicated domains more accurately than what could be achieved via traditional Riemann integration.

• The Lebesgue measure on the real line coincides with our usual understanding of length, allowing for consistent interpretations of size.
• In two or more dimensions, it measures area (in the plane) or volume (in space).

Since the Lebesgue measure possesses a countable base, it simplifies the analysis and serves as the basis for various mathematical concepts like the Lebesgue Integral.