5. For the product measure λ × λ on R2, the usual Lebesgue measure on the plane, it's very easy to show that rectangles parallel to the axes have λ × λ measure equal to their usual areas, the product of their sides. Prove this for rectangles that are not necessarily parallel to the axes.
6. Polar coordinates. Let T be the function from X := [0, ∞) × [0, 2π ) onto R2 defined by T (r,θ ) := (r · cos θ, r · sin θ ). Show that T is 1-1 on (0, ∞) × [0, 2π ). Let σ be the measure on (0, ∞) defined by σ ( A) := (A r dλ(r ) := ( 1A (r )r dλ(r ). Let µ := σ × λ on X . Show that the image measure µ ? T -1 is Lebesgue measure λ2 := λ × λ on R2. Hint: Prove λ2(B) = µ(T -1(B)) when T -1(B) is a rectangle where s r ≤ t and α θ ≤ β (B is a sector of an annulus). You might do this by calculus, or show that when (t -s)/s and β -α are small, B can be well approximated inside and out by rectangles from the last problem; then assemble small sets B to make larger ones. Show that the set of finite disjoint unions of such sets B is a ring, which generates the σ-algebra of measurable sets in R2 (use Proposition 4.1.7). (Applying a Jacobian theorem isn't allowed.)