91Ó°ÊÓ

Understanding the concepts of lim inf (limit inferior) and lim sup (limit superior) of a sequence of sets is essential in measure theory. The limit inferior, denoted as \(\liminf E_n\), represents the points that eventually remain in all sets as the sequence progresses. It is formally defined as:

• \[ \liminf E_n = \bigcup_{n=1}^{\infty} \bigcap_{k=n}^{\infty} E_k \]

This means we first take an intersection of sets starting from an index \(k\) onwards and then take their union as \(n\) starts from 1. Therefore, lim inf captures elements consistently present.

On the other hand, the limit superior \(\limsup E_n\), reveals the points that appear infinitely often in the sequence of sets. Its definition is:

• \[ \limsup E_n = \bigcap_{n=1}^{\infty} \bigcup_{k=n}^{\infty} E_k \]

This expression implies starting with a union of sets from a certain index onwards, and then finding the intersection from \(n=1\) to infinity. Hence, lim sup includes elements that recur in many sets continuously. Both of these limits help in understanding the long-term behavior of set sequences.

Continuity of measure is a fundamental property in measure theory. It allows us to understand how measures operate over unions, intersections, and limits of sequences of sets.

One of the key applications is the continuity of measure from above which deals with decreasing sequences of sets. If \(F_k\) is a descending sequence such that \( F_1 \supseteq F_2 \supseteq \ldots \), the continuity from above gives us:

• \[ \mu\left(\bigcap_{k=1}^{\infty} F_k\right) = \lim_{k \to \infty} \mu(F_k) \]

This essentially states that the measure of the intersection of the sequence is the limit of the measures of the sets involved. It is used in proving that \( \mu(\liminf E_n) \leq \liminf \mu(E_n) \).

The continuity of measure from below applies to increasing sequences where \(G_k\) is an ascending sequence such that \( G_1 \subseteq G_2 \subseteq \ldots \). For this, it says:

• \[ \mu\left(\bigcup_{k=1}^{\infty} G_k\right) = \lim_{k \to \infty} \mu(G_k) \]

This form is particularly important in discussions involving \( \limsup \), supporting proofs where it is shown that \( \limsup \mu(E_n) \leq \mu(\limsup E_n) \). Recognizing these properties enables easier manipulation of measures while working with infinite set operations.

When working in measure theory, ensuring some conditions help in applying inequalities or certain theorems correctly, one of which is the finiteness of the measure space. Specifically, the measure \( \mu(X) < \infty \) guarantees solidity in applying measure limits.

In these exercises, the condition \( \mu(X) < \infty \) became crucial for proving part (ii) of the expression, \( \limsup \mu(E_n) \leq \mu(\limsup E_n) \). Without this assumption, you may encounter discrepancies as showcased when using the counting measure on \( \mathbb{N} \). If \( X \) is infinite, taking each \( E_n = \{n, n+1, \ldots\} \), results in \( \limsup E_n = \emptyset \), making the measure \( \mu(\limsup E_n) = 0 \). However, each \( \mu(E_n) = \infty \) making \( \limsup \mu(E_n) = \infty \), leading to a contradiction.

Thus, such conditions are not merely administrative boundaries but rather pivotal elements that ensure finite results are logically consistent and align with theoretical expectations.