91Ó°ÊÓ

A countable set is a fundamental concept in mathematics, particularly in set theory and its applications. It refers to a set that has the same cardinality, or size, as the set of natural numbers. This means that there is a one-to-one correspondence between the elements of the set and the natural numbers. In simpler terms, a countable set can be thought of as a set whose elements can be listed in a sequence, even if it is an infinite sequence. For example:

• The set of all even numbers \( \{2, 4, 6, 8, \ldots\} \) is countable because we can map each even number to a natural number through the function \( f(n) = 2n \).
• The set of rational numbers is also countable, even though it might seem more complicated, they can be listed in a sequence through a more intricate pairing process.

Countable sets can be finite or infinite. However, in the context of the exercise, we focus on countably infinite sets like \( X \), where you can enumerate all its elements. This property is essential because it allows us to explore the relationship between the set and measures, such as finite or zero measures.

A \( \sigma \)-algebra is a crucial concept in measure theory, which is a branch of mathematics that studies how to assign a size or measure to different subsets of a given set. Think of a \( \sigma \)-algebra as a collection of subsets that includes the whole set, the empty set, and is closed under the operations of countable unions, countable intersections, and complements. For a set \( X \) and a \( \sigma \)-algebra \( \mathcal{A} \), it satisfies the following properties:

• \( X \in \mathcal{A} \) and \( \emptyset \in \mathcal{A} \).
• If a subset \( A \) is in \( \mathcal{A} \), then its complement \( A^c \) is also in \( \mathcal{A} \).
• The \( \sigma \)-algebra is closed under countable unions, i.e., if \( A_1, A_2, A_3, \ldots \in \mathcal{A} \), then their union \( \bigcup_{i=1}^{\infty} A_i \in \mathcal{A} \).

In the context of the exercise, \( \mathcal{A} \) represents all possible subsets of the countable set \( X \). This concept is essential because it lays the groundwork for defining and understanding measures on \( X \), especially when considering every possible subcollection of \( X \).

Countable additivity is a fundamental property that a measure must satisfy in measure theory. It essentially states that if you have an infinite collection of disjoint sets \( A_1, A_2, A_3, \ldots \), then the measure of their union is the sum of their individual measures. Formally, for a measure \( \mu \), this condition is expressed as:
\[ \mu\left(\bigcup_{i=1}^{\infty} A_i\right) = \sum_{i=1}^{\infty} \mu(A_i) \] Countable additivity ensures consistency in the measure across an infinite number of sets, making it a key part of the definition of a measure. In the context of the exercise, because each singleton subset \( \{x_i\} \) of the countable set \( X \) must be assigned a measure such that:

• The sum of the measures of these singleton sets must equal the measure of the entire set \( X \), i.e., \( \mu(X) \).
• If any of these singletons carried a positive measure, their countable sum would be infinite, contradicting the finite measure constraint.

Thus, countable additivity directly leads to the conclusion that the only viable finite measure is the trivial measure where all subsets of \( X \) have measure zero.