Question

Let \(\mu^{*}\) be an outer measure induced from a premeasure and \(\bar{\mu}\) the restriction of \(\mu^{*}\) to the \(\mu^{*}\)-measurable sets. Then \(\bar{\mu}\) is saturated.

Short answer

\(\bar{\mu}\) is saturated because subsets of any zero measure set under \(\mu^{*}\) are measurable with zero measure.

Step-by-step solution

Understanding the Problem

To show that \(\bar{\mu}\) is saturated, we need to demonstrate that for any set \(A\) such that \(\mu^{*}(A) = 0\), every subset \(B \subseteq A\) is \(\mu^{*}\)-measurable and thus has measure zero, meaning \(\bar{\mu}(B)=0\).

Recalling Definitions

A measure \(\bar{\mu}\) is saturated if for any set \(A\) with \(\mu^{*}(A)=0\), all subsets of \(A\) are \(\mu^{*}\)-measurable sets, meaning \(\bar{\mu}(B) = \mu^{*}(B) = 0\). Here, \(\mu^{*}\) is the outer measure and \(\bar{\mu}\) is the measure restricted to the \(\mu^{*}\)-measurable sets.

Applying the Carathéodory’s Criterion

A set \(B\) is \(\mu^{*}\)-measurable if for any set \(C\), \(\mu^{*}(C) = \mu^{*}(C \cap B) + \mu^{*}(C \cap B^c)\). We need to verify if subsets of a set \(A\) with \(\mu^{*}(A) = 0\) satisfy this condition.

Showing Subset Measurability for Zero Measure Sets

If \(\mu^{*}(A)=0\), then for any subset \(B\subseteq A\), we have \(\mu^{*}(B) = 0\) since the outer measure is monotonic (\(\mu^{*}(B) \leq \mu^{*}(A)\)). Thus, \(\mu^{*}(C) = \mu^{*}(C \cap B) + \mu^{*}(C \cap B^c)\) holds because both terms on the right will respect the zero-measure of \(B\).

Conclusion

Having verified that any subset \(B\) of a set \(A\) with \(\mu^{*}(A)=0\) is \(\mu^{*}\)-measurable and \(\bar{\mu}(B)=0\), we conclude that \(\bar{\mu}\) is saturated. This outcome aligns with the property of any measure that extends from an outer measure through a restriction to measurable sets.

Key concepts

Carathéodory's Criterion

Carathéodory's Criterion is a fundamental concept in measure theory that helps us determine the measurability of sets based on an outer measure. It provides a condition under which a set is considered measurable. According to Carathéodory's Criterion, a set \( B \) is \( \mu^* \)-measurable if, for any other set \( C \), the following equation holds: \[ \mu^*(C) = \mu^*(C \cap B) + \mu^*(C \cap B^c) \]This equation essentially states that the measure of any set \( C \) should be equivalent to the sum of the measures of its intersections with the set \( B \) and its complement \( B^c \).
Carathéodory’s Criterion ensures that there is no discrepancy between the parts of a set and the whole when determining measurability. This criterion is vital because it allows us to extend premeasures to \( \sigma \)-algebras, ensuring that measures assigned are consistent across all subsets considered by the outer measure.

Measurable Sets

Measurable sets are the sets for which the concept of measure is well-defined under a specific measure \( \mu \). In the context of outer measures and Carathéodory's Criterion, these are sets that satisfy Carathéodory’s condition, meaning that their presence doesn't alter the consistency of the measure across different partitions of the space.
Such sets ensure that, regardless of how we slice or partition the space, the sum of the measures of all parts matches the measure of the entire space. This property is crucial for proving many theorems in measure theory, as it secures the extension of measures from simple sets to more complex ones, enabling the development of integral calculus and probability theory. In practical terms, this means that all results applicable to measurable sets have foundation in the regularity and consistency assured by this well-defined measuring process.

Premeasure

A premeasure is a type of set function defined initially on a simpler collection of sets, called an algebra, which assigns a non-negative real number to each set in a way that is finitely additive.
Finiteness means that the measure of a finite union of disjoint sets is the sum of the measures of these sets. A premeasure might seem like a measure, but it is initially just defined on a small number of basic sets.

• Extensions: Premeasures are extended to outer measures, which cover a broader range of sets, using methods like Carathéodory's extension theorem.
• Importance: It represents a foundational step in measure theory because once extended, it allows for consistent value assignments to more complicated sets known as \( \sigma \)-algebras.
• Application: Using a premeasure, mathematicians can construct a measure that applies to more practical, complicated scenarios beyond the original limited collection of sets.

Saturation in Measure Theory

Saturation in measure theory refers to a property of a measure where, if a set has measure zero, all its subsets are also measurable and have measure zero. This concept is related to the integrity and completeness of the measure. With regards to the exercise, a measure \( \bar{\mu} \) is saturated if, for any set \( A \) such that \( \mu^*(A) = 0 \), every subset \( B \subseteq A \) is \( \mu^* \)-measurable.
Because the subsets do not add to the measure (since zero measure plus zero measure is still zero), this implies that \( \bar{\mu}(B) = 0 \) for all such subsets \( B \).

• Robustness: Saturation confirms that the measure doesn't overlook any subset of a zero-measure set.
• Consistency: It ensures that properties of zero measure sets remain intact even after taking subsets.
• Practical Implication: This is crucial when studying the properties of finer or more detailed subdivisions in measure theory.

This makes the measure \( \bar{\mu} \) comprehensive and logical when applied in continuous contexts like real-valued functions or metric spaces.