Problem 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 , Γ(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
