Self evaluation
Measures
Fingertip knowledge: What is a measure?
Five minute problem: Fix a point y ∈ R. Show that the function δy : P(R) → [0, ∞]
given by
(
1 if y ∈ A
δy (A) =
,
0 if y ∈
/A
is a measure on (R, P(R)). This measure is called the Dirac measure at the point
y.
Measurable functions
Fingertip knowledge: If (Ω, A) is a measurable space, then what does it mean for a
function f from Ω to C or R∗ to be measurable?
Five minute problem: Show that the function f : R → R given by
(
sin(x) if x > 0
f (x) =
,
χQ (x) id x ≤ 0
is measurable.
Integration of non-negative functions
Fingertip knowledge: How is the integral a non-negative function defined?
Five minute problem: Fix a point y ∈ R and consider the Dirac measure δy . Show that
Z
f dδy = f (y),
for all functions f : R → [0, ∞).
Lebesgue integration and Riemann integration
Fingertip knowledge: When do the Riemann and Lebesgue integrals agree?
Five minute problem: Compute
Z
sin(x)dλ(x).
[0,π]
1
Self evaluation
Integration of complex valued functions
Fingertip knowledge: When is a complex valued function integrable? How is the integral a real or complex values function defined?
Five minute problem: Consider the function f : (1, ∞) → R given by
f (x) =
sin(x)
.
x2
Is f integrabel?
Interchanging limits and integrals
Fingertip knowledge: Under what conditions can we interchange a limit and an integration sign? (monotone convergence and dominated convergence)
Five minute problem: Let fn (x) = sinn (x). Show
Z
lim
fn dλ = 0.
n
[0,2π]
2