Thread: Non-Lebesgue Measurable

Hi All,

I am trying to prove the following:
If B is a subset of the real numbers and satisfies lambda(B) > 0, then B includes a set that is not Lebesgue measurable.

I already know that:
1. There is a subset of (0, 1) that is not Lebesgue measurable.
2. If A is lebesgue measurable with lambda(A) > 0, then diff(A) = {x - y : x in A and y in A} includes an open interval (containing 0).

My proof so far:

By point 2. above, diff(B) contains an interval containing 0, let (-a, b) be this interval for a, b > 0. In a similar construction to point 1., (-a, b) contains a subset, F, that is not lebesgue measurable.

I am having trouble trying to work out how I can take elements from F, that are in B, and still make it non-lebesgue measurable.

Thanks.

Can anyone offer any help on this?

Thanks.