Prove something is Lebesgue measurable

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Thus, the measure of G - A has to be less than or equal to the measure of B. So, if we can show that the measure of B is zero, then we can conclude that the measure of G - A is also zero, which would imply that A is Lebesgue measurable.In summary, we can prove that A is Lebesgue measurable by using Caratheodory's Theorem and the fact that the measure of G - A is less than or equal to the measure of B, which can be shown to be zero. This implies that A is measurable, as desired.
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SNOOTCHIEBOOCHEE
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Let A be a subset of [0, 1].
And B is [0, 1] - A.
Assume m_e(A) + m_e(B) = 1.
Prove that A is Lebesgue measurable.

m_e denotes the standard outer measure.

Homework Equations



A subset E of R^n is said to be lebesgue measurable, or simply measurable, if given epsilon, there exists an open set G such that E is in G and |G - E|_e < epsilon.

The Attempt at a Solution



I'm trying to use Caratheodory's Theorem, but with no avail. I am now completley lost on this problem...

Please Reply Over!... Mike
 
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  • #2
I'd try to use the fact that G - A has to be a subset of B.
 

1. What does it mean for something to be Lebesgue measurable?

Being Lebesgue measurable means that a set or function satisfies the Lebesgue measure, which is a generalized way of measuring the size or extent of a set in mathematical analysis. It is an important concept in measure theory and allows for a more precise understanding of the properties of sets and functions.

2. How do you prove that something is Lebesgue measurable?

To prove that something is Lebesgue measurable, you must show that it satisfies the four key properties of the Lebesgue measure: non-negativity, countable additivity, translation invariance, and completeness. This can be done through various mathematical techniques, such as constructing Lebesgue outer measures, using the Carathéodory criterion, or employing the Lebesgue differentiation theorem.

3. What are some examples of sets that are Lebesgue measurable?

Some common examples of Lebesgue measurable sets include intervals, rectangles, and circles in Euclidean space. Additionally, many commonly used functions, such as continuous and differentiable functions, are also Lebesgue measurable.

4. Why is it important to prove that something is Lebesgue measurable?

Proving that something is Lebesgue measurable is important because it allows for the application of important mathematical tools and concepts, such as the Lebesgue integral, which is used to calculate the areas of irregularly shaped regions. Additionally, it helps to establish a rigorous understanding of sets and functions in measure theory and analysis.

5. Are there any alternative measures besides the Lebesgue measure?

Yes, there are alternative measures besides the Lebesgue measure, such as the Riemann measure and the Borel measure. These measures have different properties and are used in different contexts, but they are all important tools in mathematical analysis and measure theory.

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