Let E be a set which is not L-measurable,f(x)=x if x in E and f(x)=-x if x is not in E.Is f L-measurable?

Question

L-measurable means lebesgue measurable

Answer 1

Not necessarily.

Suppose we restrict ourselves to the set (0,1).

Let E be a subset of (0,1) that is not L-measurable.

Redefine the function with domain (0,1) (which is L-measurable by the way) so that:

f(x) = x if x is in E and f(x) = -x if x is in (0, 1)\E

Then we can see that the set { x in (0, 1) : f(x) > 0} = E, which is not L-measurable by assumption.

So f fails to uphold the definition of L-measurability.

Answer 2

No your function is not measurable. Because (f + Id)/x is not