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,


Let . As is a probability measure,


Let ; then

Thus, , that is


, which completes the proof.


  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


We know that is a concave function

Let , . Then

That is,

Let , ,

Then, ,

which proves the result


If , then


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


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

  1. for every , further,
  2. . . . (Minkowski's inequality)


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