Question

A measure is said to be complete if every subset of a sct of measure zero is measurable. Show that if \(A \subset \mathbb{R}\) is a set of outer measure zero, \(\mu^{*}(A)=0\), then \(A\) is Lebesgue measurable and has measur zero. Hence shew that Lebesgue measure is complcte.

Short answer

Subset \(A\) of \(\mathbb{R}\) with outer measure zero is proven to be Lebesgue measurable and to have measure zero. This allows us to show that the Lebesgue measure is complete, as per the definition of completeness, i.e., every subset of a set of measure zero is measurable.

Step-by-step solution

Prove \(A\) is Lebesgue measurable

To show \(A\) is Lebesgue measurable, first consider a set \(E\) which lies in the real numbers. We know that \(\mu^{*}(E)=\mu^{*}(E \cap A) + \mu^{*}(E \cap A^{C})\), where \(A^{C}\) represents the complement of \(A\). Given that \(\mu^{*}(A)=0\), replacing \(E \cap A\) with \(A\) gives \(\mu^{*}(E) \geq \mu^{*}(A) + \mu^{*}(E \cap A^{C})\). This implies \(\mu^{*}(E) \geq \mu^{*}(E \cap A^{C})\). Due to the sub-additivity of the outer measure, \(\mu^{*}(E) \leq \mu^{*}(E \cap A^{C})\). Therefore, \(\mu^{*}(E) = \mu^{*}(E \cap A^{C})\). This demonstrates that \(A\) is Lebesgue measurable.

Prove \(A\) has measure zero

Given that \(A\) is Lebesgue measurable, we must verify it also has measure zero. We know \(\mu^{*}(A) = 0\), and because \(A\) is measurable, its outer and inner (Lebesgue) measures are equal; i.e., \(\mu(A) = \mu^{*}(A) = 0\). Thus, \(A\) has measure zero.

Prove Lebesgue measure is complete

Finally, we need to show the Lebesgue measure is complete. Complete measure guarantees that if a set \(A\) has measure zero, every subset of \(A\) is measurable. We have proven any set \(A\) with outer measure zero is Lebesgue measurable and has measure zero. As any subset of \(A\) will also have outer (and therefore Lebesgue) measure zero, and thus will be measurable. This confirms that the Lebesgue measure is complete.

Key concepts

Measure Theory

Measure theory is a branch of mathematics that deals with the systematic way of assigning a number, known as a measure, to subsets of a given set, usually within the context of mathematics and real analysis. It allows us to generalize concepts like lengths, areas, and volumes.

In simpler terms, measure theory helps us understand how to "measure" different types of objects and shapes, even if they are highly abstract.

• Measuring lengths, areas, or volumes are examples of common measures.
• Lebesgue measure, a key concept in measure theory, is widely used to assign a measure to subsets of real numbers.
• Measure theory connects with probability, helping define probabilities in terms of measures.

This foundational theory supports various branches of mathematics, including integration and probability.

Outer Measure

The outer measure is a concept we use to define the size or extent of a set, even before determining if it's measurable within a particular system.

For any subset of the real numbers, the outer measure assigns a non-negative size, which can then be used to identify measurable sets.

• The outer measure is always sub-additive, meaning the measure of a union of sets is less than or equal to the sum of the measures of the individual sets.
• It's a step in determining whether a set is measurable according to the Lebesgue measure.
• If a set has outer measure zero, this implies it's very "small" or even "nothing" in terms of size.

Understanding outer measures is crucial for building the Lebesgue measure.

Lebesgue Measurable

A set is considered Lebesgue measurable if it can be measured using the Lebesgue measure, which relates to how well the set fits into our real number system.

This property supports the handling of more complex sets compared to previous systems, like the Riemann integral.

• To determine if a set is Lebesgue measurable, its outer measure should match its measure when intersected with any other set.
• This means the set behaves nicely with respect to the real number line.
• Having outer measure zero assures that a set is automatically Lebesgue measurable, which is central to problems involving measure zero.

This concept allows us to understand and work with sets in nuanced mathematical environments.

Complete Measure

A measure is complete if every subset of any measurable set with measure zero is also measurable.

This is an important property because it ensures the handling of all sorts of small or insignificant sets.

• The Lebesgue measure is a complete measure, meaning even the tiniest subsets of zero measure sets can be measured.
• This completeness allows for more flexibility in mathematical analysis.
• Completeness is why the Lebesgue measure is so robust in analyzing real number spaces.

Complete measures are essential for advancing in areas like real analysis and probability theory.