Download Mathematics 2224: Lebesgue integral Tutorial exercise sheet 4

Mathematics 2224: Lebesgue integral
Tutorial exercise sheet 4
1. Let f : R → [−∞, ∞] be a Lebesgue integrable function. Prove that
Z
Z
f dµ ≤
|f | dµ.
R
R
2. Let E ∈ L. Show that if f : R → [−∞, ∞] is Lebesgue integrable, then χE f is
Lebesgue integrable.
Z
Z
If f : R → [−∞, ∞] is Lebesgue integrable, let
f dµ =
χE f dµ . Show that
E
RZ
Theorem 25 holds for these integrals too (that is, show that
· dµ is linear).
E
3. Let f : R → [−∞, ∞] be a Lebesgue measurable function. Show that
Z
µ({x ∈ R : f (x) 6= 0}) = 0 =⇒
f dµ = 0
R
in each of the following cases:
(a)
(b)
(c)
(d)
f
f
f
f
= χE for some E ∈ L;
is a simple non-negative Lebesgue measurable function;
is a non-negative Lebesgue measurable function;
is a Lebesgue integrable function.
4. Let α ∈ R. Given a function f : R → [−∞, ∞], let fα : R → [∞, ∞] be the function
fα (x) = f (x − α) for x ∈ R.
First, convince yourself that the graph of fα is a horizontal translation of the graph
of f . Do you translate left or right if α > 0?
Then show that the equation
Z
Z
f dµ =
R
fα dµ
R
is satisfied in each of the following cases:
(a)
(b)
(c)
(d)
f
f
f
f
= χE for some E ∈ L;
is a simple non-negative Lebesgue measurable function;
is a non-negative Lebesgue measurable function;
is a Lebesgue integrable function.