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MATH 563, HOMEWORK
Problem 1. Suppose g : R → R is convex, which means
g(tx + (1 − t)y) ≤ tg(x) + (1 − t)g(y)
whenever x, y ∈ R and 0 ≤ t ≤ 1. Suppose f : [0, 1] → R is simple. Show that g ◦ f is simple and that
g
0
f dm
0
g ◦ f dm.
(Hint: Prove this lemma first: if
∑^ n
1
tj = 1
for some nonnegative numbers tj then
g
tj xj
tj g(xj )
for any real numbers xj .)
Problem 2. Suppose E is an open of R and E ⊆ [a, b] for some finite a and b. Show that for every > 0 there is a step function g on R and a measurable set A with
m(A) <
and such that
x /∈ A =⇒ g(x) = χE (x).
Problem 3. Suppose g is step function on R. Show that for all > 0 there is a continuous function h and a measurable set A with
m(A) <
and such that
x /∈ A =⇒ h(x) = g(x).
1
MATH 563, HOMEWORK #4 2
Definition 1. As a temporary definition, let us say that a function f : R → R is very nearly continuous if for all > 0 , there is a continuous function h : R → R and a measurable set A so that
m(A) <
and x /∈ A =⇒ h(x) = f (x).
Problem 4. Suppose that E is measurable and E ⊆ [a, b] for some finite a and b. Show that χE is very nearly continuous.
Problem 5. Show that if f and g are very nearly continuous, then so is cf + dg for any real numbers c and d.
Remark 1. We now know that all simple functions that are zero except on a finite interval are very nearly continuous.
Problem 6. Show that if f is measurable and f is zero off of [a, b] and c ≤ f (x) ≤ d for all x then for every > 0 there is a simple function g that is zero off of [a, b] with
c ≤ g(x) ≤ d
and |g(x) − f (x)| ≤
for all x.
Remark 2. All of this shows that if f is measurable, bounded and zero off a bounded set, then for every > 0 there is a set A and a continuous function g such that m(A) ≤
and x /∈ A =⇒ |g(x) − f (x)| < .
