4Inequalities and L^{p} 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.