Measure Theory/L^p spaces

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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]

  1. Putting ,
  1. 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,

  1. for every , further,
  2. . . . (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