Set 15 1. Let Zi RSN , i = 1, . . . , n be a collection of zero measure sets. Show that the union Zi also has zero measure. 2. Let f : [a, b] R be a bounded, integrable function. (a) Show that the graph of f , (f ) := {(x, f (x)) : x [a, b]} R2 has zero content. (b) If f is non-negative, show that S = {(x, y) : a x b, 0 y f (x)} Rb is measurable, and m(S) = a f (x)dx. 3. Show that if f : R R2 is a C 1 function, then for any interval I R, f (I) has zero Jordan measure. 4. If S = {x1 , . . . , xn } is a finite set consisting of precisely n-elements, show that S has zero Jordan measure. 5. Let f : [a, b] R be a Riemann integrable function. If g : [a, b] R is another function and S = {x : f (x) 6= g(x)} contains exactly n-points, show that g is also Riemann integrable. [Note: You must prove this from scratch. If you wish to invoke a corollary or result from class, you must first prove it.] 1