Prove something is Lebesgue measurable

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To prove that the subset A of [0, 1] is Lebesgue measurable given that m_e(A) + m_e(B) = 1, one can utilize the definition of Lebesgue measurability, which requires finding an open set G that contains A with a small measure difference. The discussion highlights an attempt to apply Caratheodory's Theorem, though the user expresses difficulty in progressing with the solution. Another participant suggests leveraging the relationship between the open set G and the complement B, indicating that G - A must be a subset of B. The conversation emphasizes the need for a clear application of measure theory principles to establish the measurability of A.
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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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I'd try to use the fact that G - A has to be a subset of B.
 

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