Does there exists a set which has the exterior measure,but doesn't Lebesgue measurable? Does there exists a set which has the exterior measure,but doesn't Lebesgue measurable?
Can one give a example of this ?
I do really puzzled.

Every set in $\mathbb R^n$ has an exterior measure (sometimes called outer measure).
So every subset that is not Lebesgue measurable would be an such an example which you are looking for.

Does there exists a set which has the exterior measure,but doesn't Lebesgue measurable?
  Since there exist subsets that are not Lebesgue measurable, the answer is yes.

Note that the existence of sets that are not Lebesgue measurable is not constructive. Searching for "Vitali sets" will give more information about this.
