Prove something is Lebesgue measurable

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SUMMARY

To prove that a subset A of [0, 1] is Lebesgue measurable given that m_e(A) + m_e(B) = 1, where B is the complement of A in [0, 1], one can utilize the definition of Lebesgue measurability. Specifically, a set E is Lebesgue measurable if for any epsilon, there exists an open set G such that E is contained in G and the outer measure of the difference |G - E|_e is less than epsilon. The discussion highlights the application of Caratheodory's Theorem as a potential method for the proof.

PREREQUISITES
  • Understanding of Lebesgue measure and outer measure (m_e).
  • Familiarity with the definition of Lebesgue measurable sets.
  • Knowledge of Caratheodory's Theorem.
  • Basic concepts of set theory and open sets in R^n.
NEXT STEPS
  • Study the application of Caratheodory's Theorem in proving measurability.
  • Explore the properties of Lebesgue outer measure and its implications.
  • Investigate examples of Lebesgue measurable and non-measurable sets.
  • Learn about the construction of open sets in R^n and their relationship to measurable sets.
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Mathematicians, students studying real analysis, and anyone interested in measure theory and its applications in mathematical proofs.

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
 
Last edited:
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I'd try to use the fact that G - A has to be a subset of B.
 

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