4Inequalities and Lp spaces
II Probability and Measure
4 Inequalities and L
p
spaces
Eventually, we will want to define the L
p
spaces as follows:
Definition
(
L
p
spaces)
.
Let (
E, E, µ
) be a measurable space. For 1
≤ p < ∞
,
we define
L
p
=
L
p
(
E, E, µ
) to be the set of all measurable functions
f
such that
kfk
p
=
Z
|f|
p
dµ
1/p
< ∞.
For p = ∞, we let L
∞
= L
∞
(E, E, µ) to be the space of functions with
kfk
∞
= inf{λ ≥ 0 : |f| ≤ λ a.e.} < ∞.
However, it is not clear that this is a norm. First of all,
kfk
p
= 0 does not
imply that
f
= 0. It only means that
f
= 0 a.e. But this is easy to solve. We
simply quotient out the vector space by functions that differ on a set of measure
zero. The more serious problem is that we don’t know how to prove the triangle
inequality.
To do so, we are going to prove some inequalities. Apart from enabling us to
show that
k · k
p
is indeed a norm, they will also be very helpful in the future
when we want to bound integrals.