Need help with problem 8, Real analysis and measure theory course.

8. (The first part of this problem is somewhat difficult.) Recall the setting of an earlier homework exercise: Let A and B be Borel subsets of R, each having positive and finite Lebesgue measure. It is a fact that the function f: R x R - [0, co) defined by f(x, y) = XA(y - x)XB(y) is measurable with respect to the product o-algebra B[R] & BIR]. The conclusions were: f(x, y) = Xx+A(y)XB(y), and if we define g(x) = fr f(x, y) dx(y), then g(x) = [(x + A) n B| and fig d) = |A| |B). Now prove the following: (a) The function g is continuous. Suggestion: Either approximate A by a finite disjoint union of open intervals or approximate XA in L' by a continuous function that equals 0 outside some bounded set. (b) The set B - A = {b - a: a E A, bE B} contains a nonempty open interval