91Ó°ÊÓ

A measurable space is a fundamental concept in measure theory. It consists of a set, often denoted by \(X\), paired with a \(\sigma\)-algebra \(\delta\). The \(\sigma\)-algebra is a collection of subsets of \(X\) that includes the empty set and is closed under the operations of complement and countable unions. This means that if you have a sequence of sets within the \(\sigma\)-algebra, their union will also be a part of it.

Measurable spaces help provide a framework for defining and analyzing measures. In our exercise, \(\llbracket X, \delta \rrbracket\) represents such a measurable space, giving context to where the measures \(\mu_n\) act. Knowing whether a set is in \(\delta\) determines if a measure can be assigned to it. Thus, understanding measurable spaces is crucial for dealing with measures and integrals over these spaces.

In measure theory, a sequence of measures \(\left(\mu_n\right)\) is a collection of measure functions that are indexed by integers \(n\). These measures are defined on the same \(\sigma\)-algebra. In our exercise context, each measure \(\mu_n\) is assigned to the elements of \(\delta\), the \(\sigma\)-algebra of the measurable space.

An important property of this sequence in the exercise is monotonicity, meaning that for any set \(E\) in \(\delta\), \(\mu_{n+1}(E) \geq \mu_n(E)\). This non-decreasing nature ensures that as \(n\) increases, the measure assigned to \(E\) does not decrease, which lays the groundwork for defining a limiting behavior of these measures, crucial for proving the measure \(\mu(E) = \lim \mu_n(E)\) is well-defined.

The Monotone Convergence Theorem (MCT) is a key result in measure theory that connects sequences of functions and their integrals. It states that if a sequence of non-negative measurable functions \(f_n\) increases pointwise to a function \(f\), then the integral of \(f_n\) over a measure \(\mu\) converges to the integral of \(f\).

In the context of our exercise, MCT is invoked to show that the limit of the integrals of a function \(f\) with respect to the measures \(\mu_n\) equals the integral of \(f\) with respect to the limiting measure \(\mu\). Because \((\mu_n)\) is an increasing sequence of measures, it ensures that \(\int f \, d\mu = \lim \int f \, d\mu_n\), asserting the convergent behaviour required to transition from the measures \(\mu_n\) to \(\mu\).

This theorem is powerful because it provides conditions under which taking limits and integrals can be interchanged, which is not always safe otherwise.

Countable additivity is a fundamental property that every measure must satisfy. It asserts that if you have a countable collection of disjoint sets \((E_i)\), then the measure of their union is the sum of the measures of the individual sets: \(\mu\left(\bigcup_{i=1}^\infty E_i\right) = \sum_{i=1}^\infty \mu(E_i)\). This property is crucial in ensuring that measures behave predictably when you combine multiple subsets of a space.

In our exercise, we demonstrate that the limit measure \(\mu\) inherits the countable additivity property from the sequence \((\mu_n)\). Since each \(\mu_n\) is itself a measure, and assuming the measure sequence \(\mu_n\) is increasing, we can take limits on both sides of the equality to verify that \(\mu\) remains countably additive. This establishes \(\mu\) as a legitimate measure by confirming that it obeys this essential measure property.