Measure, Integration & Real Analysis, 2021a

Section 2E Convergence of Measurable Functions 71
EXERCISES 2E
1 Suppose X is a finite set. Explain why a sequence of functions from X to R that
converges pointwise on X also converges uniformly on X.
2 Give an example of a sequence of functions from Z + to R that converges
pointwise on Z + but does not converge uniformly on Z + .
3 Give an example of a sequence of continuous functions f 1 , f 2 ,... from [0, 1] to
R that converges pointwise to a function f : [0, 1] → R that is not a bounded
function.
4 Prove or give a counterexample: If A ⊂ R and f 1 , f 2 ,... is a sequence of
uniformly continuous functions from A to R that converges uniformly to a
function f : A → R, then f is uniformly continuous on A.
5 Give an example to show that Egorov’s Theorem can fail without the hypothesis
that μ(X) < ∞.
6 Suppose (X, S, μ) is a measure space with μ(X) < ∞. Suppose f 1 , f 2 ,...is a
sequence of S-measurable functions from X to R such that lim k→∞ f k (x) =∞
for each x ∈ X. Prove that for every ε > 0, there exists a set E ∈Ssuch that
μ(X \ E) < ε and f 1 , f 2 ,...converges uniformly to ∞ on E (meaning that for
every t > 0, there exists n ∈ Z + such that f k (x) > t for all integers k ≥ n and
all x ∈ E).
[The exercise above is an Egorov-type theorem for sequences of functions that
converge pointwise to ∞.]
7 Suppose F is a closed bounded subset of R and g 1 , g 2 ,... is an increasing
sequence of continuous real-valued functions on F (thus g 1 (x) ≤ g 2 (x) ≤···
for all x ∈ F) such that sup{g 1 (x), g 2 (x),...} < ∞ for each x ∈ F. Define a
real-valued function g on F by
g(x) = lim
k→∞
g k (x).
Prove that g is continuous on F if and only if g 1 , g 2 ,...converges uniformly on
F to g.
[The result above is called Dini’s Theorem.]
8 Suppose μ is the measure on (Z + ,2 Z+ ) defined by
1
μ(E) = ∑
2
n∈E
n .
Prove that for every ε > 0, there exists a set E ⊂ Z + with μ(Z + \ E) < ε
such that f 1 , f 2 ,...converges uniformly on E for every sequence of functions
f 1 , f 2 ,... from Z + to R that converges pointwise on Z + .
[This result does not follow from Egorov’s Theorem because here we are asking
for E to depend only on ε. In Egorov’s Theorem, E depends on ε and on the
sequence f 1 , f 2 ,....]
Measure, Integration & Real Analysis, by Sheldon Axler