Recall that an space is defined as
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.
- Putting ,
- If is finite, is a counting measure, and if , then
For every , define
Let such that . Let and .
We know that is a concave function
Let , . Then
Let , ,
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,
- for every , further,
- . . . (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