Measure, Integration & Real Analysis, 2021a

Section 3A Integration with Respect to a Measure 85
4 Give an example of a Borel measurable function f : [0, 1] → (0, ∞) such that
L( f , [0, 1]) = 0.
[Recall that L( f , [0, 1]) denotes the lower Riemann integral, which was defined
in Section 1A. Ifλ is Lebesgue measure on [0, 1], then the previous exercise
states that ∫ f dλ > 0 for this function f , which is what we expect of a positive
function. Thus even though both L( f , [0, 1]) and ∫ f dλ are defined by taking
the supremum of approximations from below, Lebesgue measure captures the
right behavior for this function f and the lower Riemann integral does not.]
5 Verify the assertion that integration with respect to counting measure is summation
(Example 3.6).
6 Suppose (X, S, μ) is a measure space, f : X → [0, ∞] is S-measurable, and P
and P ′ are S-partitions of X such that each set in P ′ is contained in some set in
P. Prove that L( f , P) ≤L( f , P ′ ).
7 Suppose X is a set, S is the σ-algebra of all subsets of X, and w : X → [0, ∞]
is a function. Define a measure μ on (X, S) by
μ(E) = ∑ w(x)
x∈E
for E ⊂ X. Prove that if f : X → [0, ∞] is a function, then
∫
f dμ = ∑ w(x) f (x),
x∈X
where the infinite sums above are defined as the supremum of all sums over
finite subsets of E (first sum) or X (second sum).
8 Suppose λ denotes Lebesgue measure on R. Give an example of a sequence
f 1 , f 2 ,... of simple Borel measurable functions from R to [0, ∞) such that
lim k→∞ f k (x) =0 for every x ∈ R but lim k→∞
∫
fk dλ = 1.
9 Suppose μ is a measure on a measurable space (X, S) and f : X → [0, ∞] is an
S-measurable function. Define ν : S→[0, ∞] by
∫
ν(A) = χ A
f dμ
for A ∈S. Prove that ν is a measure on (X, S).
10 Suppose (X, S, μ) is a measure space and f 1 , f 2 ,...is a sequence of nonnegative
S-measurable functions. Define f : X → [0, ∞] by f (x) =∑ ∞ k=1 f k(x). Prove
that
∫
∞ ∫
f dμ = ∑ f k dμ.
k=1
11 Suppose (X, S, μ) is a measure space and f 1 , f 2 ,...are S-measurable functions
from X to R such that ∑ ∞ ∫
k=1 | fk | dμ < ∞. Prove that there exists E ∈Ssuch
that μ(X \ E) =0 and lim k→∞ f k (x) =0 for every x ∈ E.
Measure, Integration & Real Analysis, by Sheldon Axler