Thread: Dense

Let A be a bounded measurable subset of R, and let B be a proper measurable subset of A such that m(B)=m(A). Prove that B is dense in A.

Suppose that B is not dense in A. Then there is a point and a ball U around x which meets no point of B. Since B is contained in A-U, . Now , but m(U) must be positive, a contradiction.