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,



Let  . As   is a probability measure,  


Let  ; then  

Thus,  , that is  


 , 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



We know that   is a concave function

Let  ,  . Then  

That is,  

Let  ,  ,  


Then,  ,

which proves the result

Corollary edit

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  

Theorem edit

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

  1.   for every  , further,  
  3.   . . . (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