# Lebesgue Outer Measure .... Carothers, Proposition 16.2 (i) ....

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In summary: Your Name]In summary, I provided a formal and rigorous proof for Proposition 16.2 (i) in Carothers' book on Real Analysis, which states that the Lebesgue outer measure of any set E is always non-negative. This proof involves using the definition of Lebesgue outer measure and assuming for contradiction that m*(E) < 0, then showing that this leads to a contradiction.
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I am reading N. L. Carothers' book: "Real Analysis". ... ...

I am focused on Chapter 16: Lebesgue Measure ... ...

I need help with the proof of Proposition 16.2 part (i) ...

Proposition 16.2 and its proof read as follows:

Carothers does not prove Proposition 16.2 (i) above ...

Although it seems intuitively obvious, I am unable to construct and express a valid, convincing, formal and rigorous proof of the result ...

Can someone please demonstrate a formal and rigorous proof of Proposition 16.2 (i) above ...

Peter
========================================================================================================It may help readers of the above post to have access to Carothers introduction to Lebesgue outer measure ... so I am providing the same as follows:
Hope that helps ...

Peter

Last edited:
Dear Peter,

Thank you for reaching out for help with Proposition 16.2 (i) in Carothers' book. I am happy to assist you in constructing a formal and rigorous proof for this proposition.

First, let's review the definition of Lebesgue outer measure, which is denoted by m*(E) for a set E. The Lebesgue outer measure of a set E is defined as the infimum of the sum of the lengths of intervals that cover E. In other words:

m*(E) = inf{∑(b-a) | E ⊆ ∪(a,b)}

where a and b are real numbers representing the endpoints of intervals that cover E.

Now, let's move on to the proof of Proposition 16.2 (i):

Proposition 16.2 (i): For any set E, m*(E) ≥ 0.

Proof:

Assume for contradiction that m*(E) < 0. Then, by definition, there exists a sequence of intervals {I_n} such that E ⊆ ∪(a_n, b_n) and ∑(b_n - a_n) < 0. This implies that ∑(b_n - a_n) is a negative number, which is a contradiction since the sum of lengths of intervals should always be non-negative.

Hence, our assumption that m*(E) < 0 must be false, and therefore m*(E) ≥ 0.

I hope this proof helps you to understand and construct your own formal and rigorous proof for Proposition 16.2 (i). If you have any further questions, please do not hesitate to ask.

## 1. What is Lebesgue Outer Measure?

Lebesgue Outer Measure is a mathematical concept used in measure theory to assign a numerical value to sets of points in a given space. It is a generalization of the concept of length, area, and volume to higher dimensions.

## 2. How is Lebesgue Outer Measure calculated?

The Lebesgue Outer Measure of a set A is calculated by taking the infimum (greatest lower bound) of the sum of the lengths of intervals that cover the set A. In other words, it is the smallest possible number that can be used to cover the set A.

## 3. What is Carothers' Proposition 16.2 (i)?

Carothers' Proposition 16.2 (i) is a statement in measure theory that states that the Lebesgue Outer Measure of a countable union of sets is less than or equal to the sum of the Lebesgue Outer Measures of each individual set. In other words, the Lebesgue Outer Measure is subadditive.

## 4. How is Carothers' Proposition 16.2 (i) useful?

Carothers' Proposition 16.2 (i) is useful because it allows us to calculate the Lebesgue Outer Measure of a countable union of sets by simply adding the individual Lebesgue Outer Measures together. This simplifies the calculation process and makes it easier to work with in measure theory.

## 5. What are some applications of Lebesgue Outer Measure?

Lebesgue Outer Measure has many applications in mathematics, particularly in measure theory and analysis. It is used to define the Lebesgue measure, which is a more general and powerful measure than the traditional concept of length, area, and volume. It also has applications in probability theory, where it is used to define the probability of events in a given space.

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