Prove something is Lebesgue measurable

[SNOOTCHIEBOOCHEE]

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

 

[EnumaElish]

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