91Ó°ÊÓ

In measure theory, the concept of a decreasing sequence of measures arises when considering a set of measures that steadily reduces. This is formalized as a sequence \(\{\mu_n\}\) where each measure \(\mu_n\) is greater than or equal to the next, \(\mu_{n+1}\), for all \(n\). This means, for each measurable set \(E\), the values \(\mu_n(E)\) form a non-increasing sequence.

Such a sequence is particularly useful because it ensures the existence of a limit for \(\mu(E) = \lim_{n \to \infty} \mu_n(E)\) as \(n\) becomes very large. Due to the property that \(\mu_1(X) < \infty\), the sequence \(\mu_n(E)\) doesn't drop to negative infinity, and hence converges to a finite value.

Understanding decreasing sequences is pivotal as it provides a framework to define new measures by taking limits of existing non-increasing ones.

Every measure, \(\mu_n\), in a sequence is equipped with the property of being non-negative, meaning for any measurable set \(E\), \(\mu_n(E) \geq 0\). This derives from the basic axioms of measure theory, ensuring that measures can be thought of as generalized notions of 'size' or 'volume', which are inherently non-negative.

When you take the limit of a sequence of non-negative numbers, the resulting limit will also be non-negative. Hence, for our limit measure \(\mu(E) = \lim_{n \to \infty} \mu_n(E)\), non-negativity is preserved, with \(\mu(E) \geq 0\) for all measurable sets \(E\).

The preservation of non-negativity when transitioning from \(\mu_n\) measures to the limit measure \(\mu\) underpins a key property that all resulting measures must satisfy.

The null set property is a straightforward yet crucial characteristic in measure theory. It asserts that any measure assigns a zero value to the empty set. For our sequence of measures \(\mu_n\), this property means \(\mu_n(\emptyset) = 0\) for every \(n\).

When considering the limit measure \(\mu\), constructed from \(\mu_n\), this null set property continues to hold. By taking the limit, we find \(\mu(\emptyset) = \lim_{n \to \infty} \mu_n(\emptyset) = 0\).

Ensuring this property holds is vital as it maintains consistency in the framework of measure theory; no measure can attribute anything greater than zero size to a non-existent set.

Countable additivity is one of the core axioms of measures. It tells us that for a countable collection of disjoint measurable sets, the measure of their union is the sum of their individual measures.

In a formal setting, if \(\{E_i\}_{i=1}^{\infty}\) is a collection of disjoint sets in \(\mathcal{S}\), then for any measure \(\mu_n\), we have \(\mu_n(\bigcup_{i=1}^{\infty} E_i) = \sum_{i=1}^{\infty} \mu_n(E_i)\).

This property follows naturally to the limit measure \(\mu\) as well. Through techniques such as the Monotone Convergence Theorem, we can prove that \(\mu(\bigcup_{i=1}^{\infty} E_i) = \sum_{i=1}^{\infty} \mu(E_i)\), thus maintaining countable additivity.

This characteristic ensures that adding up "pieces" of a measurable set to find the total measure is always consistent, which is foundational to the measure's ability to assess 'size' appropriately.