# Measure Theory/L^p spaces

Recall that an space is defined as

## Jensen's inequality[edit]

Let be a probability measure space.

Let , be such that there exist with

If is a convex function on then,

**Proof**

Let . As is a probability measure,

Let

Let ; then

Thus, , that is

Put

, which completes the proof.

### Corollary[edit]

- Putting ,

- If is finite, is a counting measure, and if , then

For every , define

## Holder's inequality[edit]

Let such that . Let and .

Then, and

**Proof**

We know that is a concave function

Let , . Then

That is,

Let , ,

Then, ,

which proves the result

### Corollary[edit]

If , then

**Proof**

Let , ,

Then, , and hence

We say that if , *almost everywhere* on if . Observe that this is an equivalence relation on

If is a measure space, define the space to be the set of all equivalence classes of functions in

## Theorem[edit]

The space with the norm is a normed linear space, that is,

- for every , further,
- . . . (Minkowski's inequality)

**Proof**

1. and 2. are clear, so we prove only 3. The cases and (see below) are obvious, so assume that and let be given. Hölder's inequality yields the following, where is chosen such that so that :

Moreover, as is convex for ,

This shows that so that we may divide by it in the previous calculation to obtain .

Define the space . Further, for define