Question

Let \((X, \mathcal{M}, \mu)\) be a finite measure space. a. If \(E, F \in \mathcal{M}\) and \(\mu(E \Delta F)=0\), then \(\mu(E)=\mu(F)\). b. Say that \(E \sim F\) if \(\mu(E \Delta F)=0\); then \(\sim\) is an equivalence relation on \(\mathcal{M}\). c. For \(E, F \in \mathcal{M}\), define \(\rho(E, F)=\mu(E \Delta F)\). Then \(\rho(E, G) \leq \rho(E, F)+\) \(\rho(F, G)\), and hence \(\rho\) defines a metric on the space \(\mathcal{M} / \sim\) of equivalence classes.

Short answer

(a) \(\mu(E) = \mu(F)\); (b) \(\sim\) is an equivalence relation; (c) \(\rho\) is a metric on \(\mathcal{M} / \sim\).

Step-by-step solution

Understanding the Problem Statement

We are given a finite measure space \((X, \mathcal{M}, \mu)\), where \(X\) is a set, \(\mathcal{M}\) is a \(\sigma\)-algebra of subsets of \(X\), and \(\mu\) is a measure such that \(\mu(X) < \infty\). We have to prove three claims regarding symmetric difference of sets and properties of a relation and metric defined on these sets.

Proving Part (a): Property of Measure with Symmetric Difference

Given \(E, F \in \mathcal{M}\) such that \(\mu(E \Delta F) = 0\). By the definition of symmetric difference, \(E \Delta F = (E \setminus F) \cup (F \setminus E)\). Since the measure of the union is zero, both \(\mu(E \setminus F) = 0\) and \(\mu(F \setminus E) = 0\). Therefore, \(\mu(E) = \mu((E \setminus F) \cup (E \cap F)) = \mu(E \cap F)\) and similarly, \(\mu(F) = \mu(F \cap E)\). Thus, \(\mu(E) = \mu(F)\).

Proving Part (b): The Relation is an Equivalence

Let's verify that \(E \sim F\) is an equivalence relation. - **Reflexivity**: \(E \Delta E = \emptyset\), so \(\mu(E \Delta E) = 0\). Thus, \(E \sim E\).- **Symmetry**: If \(E \sim F\), then \(\mu(E \Delta F) = 0\). Since symmetric difference is symmetric, \(\mu(F \Delta E) = 0\). Thus, \(F \sim E\).- **Transitivity**: If \(E \sim F\) and \(F \sim G\), then \(\mu(E \Delta F) = 0\) and \(\mu(F \Delta G) = 0\). We need to show \(\mu(E \Delta G) = 0\). With \(E \Delta G = (E \Delta F) \Delta (F \Delta G)\), and using properties of measure, \(\mu((E \Delta F) \Delta (F \Delta G)) = 0\). Hence, \(E \sim G\). This confirms \(\sim\) is an equivalence relation.

Proving Part (c): Metric Properties of \(\rho\)

We need to show that \(\rho(E, G) \leq \rho(E, F) + \rho(F, G)\), where \(\rho(E, F) = \mu(E \Delta F)\). By subadditivity of the measure and properties of symmetric difference:- \(E \Delta G = (E \Delta F) \Delta (F \Delta G)\) implies \[ \mu(E \Delta G) \leq \mu((E \Delta F) \cup (F \Delta G)) \leq \mu(E \Delta F) + \mu(F \Delta G) \].This shows that \(\rho\) satisfies the triangle inequality.- Additionally, \(\rho(E, F) = 0\) if and only if \(\mu(E \Delta F) = 0\). This implies \(E \sim F\) and confirms \(\rho\) is a metric on \(\mathcal{M} / \sim\).

Key concepts

Finite Measure Space

Let's begin exploring what a finite measure space is. A finite measure space is represented by the triplet \((X, \mathcal{M}, \mu)\). Here, each component has a specific role:

• \(X\) is a set, often referred to as the underlying set or space, containing all possible elements we are considering.
• \(\mathcal{M}\) is a \(\sigma\)-algebra, which is essentially a collection of subsets of \(X\) that satisfies certain properties, such as being closed under complementation and countable unions.
• \(\mu\) is the measure, a function that assigns a non-negative real number or infinity, \(\infty\), to each set in \(\mathcal{M}\), representing the "size" or "volume" of a set.

In a finite measure space, the measure of the entire space \(\mu(X)\) is less than infinity, meaning the total measure is bounded.
This ensures that there is no infinite "size" within the elements we are measuring, which helps simplify analysis and problem-solving in measure theory.

Symmetric Difference

The concept of symmetric difference is vital when comparing two sets. Suppose you have two sets, \(E\) and \(F\). Their symmetric difference, denoted \(E \Delta F\), is defined as the set of elements that are in either of the sets but not in their intersection.
This can be expressed mathematically as:\[E \Delta F = (E \setminus F) \cup (F \setminus E)\]Here:

• \(E \setminus F\) is the set of elements in \(E\) but not in \(F\).
• \(F \setminus E\) is the set of elements in \(F\) but not in \(E\).

The symmetric difference basically filters out the common elements between \(E\) and \(F\) and only keeps the unique elements from each set.
In measure theory, if the measure of the symmetric difference \(\mu(E \Delta F) = 0\), it implies that for all practical purposes, the sets \(E\) and \(F\) are equivalent in the space being measured, as the difference is unmeasurable (effectively zero).

Equivalence Relation

Equivalence relations are a way to group objects that share a specific property or relation. For a relation \(\sim\) on a set to be considered an equivalence relation, it must satisfy three properties:

• Reflexivity: For any element \(E\) in the set, \(E \sim E\) should hold true.
• Symmetry: If \(E \sim F\), then \(F \sim E\) must also hold true.
• Transitivity: If \(E \sim F\) and \(F \sim G\), then it must follow that \(E \sim G\).

In the context of measure spaces, the relation \(E \sim F\) defined by \(\mu(E \Delta F) = 0\) becomes an equivalence relation here. This means that sets that are indistinguishable by measure (i.e., their symmetric difference has a measure of zero) are grouped together into equivalence classes.
This is incredibly useful in mathematics as it allows us to treat sets that differ only in "negligible" ways as being essentially the same, simplifying analysis and calculation.

Metric Spaces

To understand how metric spaces come into play, let's explore the notion of a metric. A metric on a set is essentially a function that defines the distance between elements of that set. For this to be valid, the function must adhere to certain properties:

• **Non-negativity:** The distance \(\rho(E, F) \geq 0\) for any sets \(E\) and \(F\), and \(\rho(E, F) = 0\) if and only if \(E \sim F\).
• **Symmetry:** The distance is symmetric, meaning \(\rho(E, F) = \rho(F, E)\) for any sets \(E\) and \(F\).
• **Triangle Inequality:** For any sets \(E\), \(F\), and \(G\), the inequality \(\rho(E, G) \leq \rho(E, F) + \rho(F, G)\) must hold.

In the context of measure theory, the metric \(\rho(E, F) = \mu(E \Delta F)\) represents how "different" two sets are based on the measure of their symmetric difference.
This distance function creates a metric space on the equivalence classes formed by \(\sim\). These metric properties ensure that the logical consistency needed for analysis is preserved, allowing mathematicians to properly quantify and compare differences between equivalence classes.