Question

Let \(\mu\) be a finite measure on \((X, \mathcal{M})\), and let \(\mu^{*}\) be the outer measure induced by \(\mu\). Suppose that \(E \subset X\) satisfies \(\mu^{*}(E)=\mu^{*}(X)\) (but not that \(E \in \mathcal{M}\) ). a. If \(A, B \in \mathcal{M}\) and \(A \cap E=B \cap E\), then \(\mu(A)=\mu(B)\). b. Let \(\mathcal{M}_{E}=\\{A \cap E: A \in \mathcal{M}\\}\), and define the function \(\nu\) on \(\mathcal{M}_{E}\) defined by \(\nu(A \cap E)=\mu(A)\) (which makes sense by (a)). Then \(\mathcal{M}_{E}\) is a \(\sigma\)-algebra on \(E\) and \(\nu\) is a measure on \(\mathrm{M}_{E}\).

Short answer

a) \(\mu(A) = \mu(B)\). b) \(\mathcal{M}_{E}\) is a \(\sigma\)-algebra on \(E\), and \(\nu\) is a measure.

Step-by-step solution

Understanding the Outer Measure

The outer measure \(\mu^{*}\) is defined such that for any subset \(E\) of \(X\), \(\mu^{*}(E)\) is the infimum of the sum of measures of countable collections of measurable sets that cover \(E\). Here, we know \(\mu^{*}(E)=\mu^{*}(X)\), which implies that \(E\) covers \(X\) up to a set with \(\mu^{*}\) measure zero.

Establishing \(\mu(A) = \mu(B)\) given \(A \cap E = B \cap E\)

Given that \(A, B \in \mathcal{M}\) and \(A \cap E = B \cap E\), the sets differ only by elements not in \(E\). Since \(\mu^{*}(E) = \mu^{*}(X)\), any set not in \(E\) has measure zero. Therefore, \(\mu(A \setminus E) = \mu(B \setminus E) = 0\), implying \(\mu(A) = \mu(B)\).

Defining the Set \(\mathcal{M}_{E}\) and Measure \(\nu\)

\(\mathcal{M}_{E} = \{A \cap E : A \in \mathcal{M}\}\) is the collection of intersections of measurable sets with \(E\). The function \(u(A \cap E) = \mu(A)\) makes sense because, by Step 2, \(\mu(A)\) only depends on \(A \cap E\).

Proving \(\mathcal{M}_{E}\) is a \(\sigma\)-algebra

To prove \(\mathcal{M}_{E}\) is a \(\sigma\)-algebra, show that it is closed under complementation and countable unions:1. If \(A \cap E \in \mathcal{M}_{E}\), then \( (E \setminus (A \cap E)) = (E \cap (X \setminus A)) \in \mathcal{M}_{E}\), as \((X \setminus A) \in \mathcal{M}\).2. If \(A_i \cap E \in \mathcal{M}_{E}\) for each \(i\), then \(\bigcup A_i \cap E = (\bigcup A_i) \cap E \in \mathcal{M}_{E}\) since \(\bigcup A_i\) is measurable.

Verifying \(\nu\) is a Measure on \(\mathcal{M}_{E}\)

To verify \(u\) is a measure on \(\mathcal{M}_{E}\), verify countable additivity:1. If \(A_i \cap E\) are disjoint, \( u\left( \bigcup (A_i \cap E) \right) = u\left( \bigcup A_i \cap E \right) = \mu\left( \bigcup A_i \right)\).2. Since \(\mu\) is a measure, \(\mu\left( \bigcup A_i \right) = \sum \mu(A_i)\), leading to \(u\left( \bigcup (A_i \cap E) \right) = \sum u(A_i \cap E)\).

Key concepts

Finite Measure

A finite measure is a type of mathematical measurement that assigns a real non-negative number to each set within a given collection, known as a sigma-algebra. This measure is called 'finite' because the measure of the entire space it is defined on is a finite number. The importance of a finite measure comes in its application:

• It simplifies calculations by ensuring the measure doesn’t go to infinity, keeping all assessed values within a manageable range.
• This constraint allows for more straightforward comparisons between different sets within the sigma-algebra.
• It guarantees certain properties, such as countable additivity, hold true.

In the context of the given exercise, \(\mu\) is a finite measure on a measurable space \( (X, \mathcal{M}) \), ensuring that the set we're working with has a measure that doesn’t exceed a finite limit.

Sigma-Algebra

A sigma-algebra is a mathematical structure that is foundational to measure theory. It is a collection of sets that satisfy specific properties:

• It includes the entire space and the empty set.
• It is closed under the operation of complementing sets.
• It is closed under countable unions (and thus countable intersections due to De Morgan’s laws).

These properties make sigma-algebras essential for defining measures, as they provide the necessary framework that measures can be applied consistently across arbitrary collections of sets.
In this exercise, \(\mathcal{M}\) is a sigma-algebra over the space \(X\), and the set \(\mathcal{M}_{E}\) is established as a sigma-algebra specific to a subset \(E\). This is done by considering intersections of measurable sets in \(\mathcal{M}\) with \(E\), thus maintaining all the necessary properties that characterize sigma-algebras.

Measure Theory

Measure theory is the mathematical study centered around the quantitative assessment of objects in space typically using various forms of measures. It generalizes the notion of length, area, and volume:

• It formalizes the process of measuring complex and abstract spaces where traditional methods fall short.
• One of its primary goals is to provide a foundation for integration, probability, and related subjects.

The core principles in measure theory include the concepts of measurable space, sigma-algebra, and measurable sets. The key feature of measure theory ensures that sets can be assigned a measure systematically, aligning with our intuitions of 'size'.
In the exercise, we apply these principles by defining a measure \(u\) according to results from classical measures \(\mu\) on the intersections with \(E\), demonstrating one function of measure theory is the translation of abstract ideas to concrete values that align with intuitive concepts of size and coverage.

Measurable Sets

Measurable sets are central to measure theory. They are parts of space's sigma-algebra, meaning they satisfy the properties required for a set to be assigned a measure. Understanding measurable sets involves realizing:

• They are the building blocks upon which measures are constructed.
• They allow us to meaningfully talk about the "size" or "volume" of different parts of a space.

In the realm of the given exercise, the operations performed on the set \(E\) relative to measurable sets like \(A\) and \(B\) demonstrate how we can identify and isolate particular parts of \(E\) in the context of the larger measurable space. This helps ensure that logical operations and manipulations will preserve the meaningfulness and applicability of measures, allowing for substantive calculations and comparisons. Therefore, allowing \( A \cap E \) for \( A \in \mathcal{M} \) to remain within the sphere of measurable sets.