Measure Theory/L^p spaces

Recall that an space is defined as

Jensen's inequalityEdit

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.

CorollaryEdit

  1. Putting  ,
     
  1. If   is finite,   is a counting measure, and if  , then
     

For every  , define  

Holder's inequalityEdit

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

CorollaryEdit

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  

TheoremEdit

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

  1.   for every  , further,  
  2.  
  3.   . . . (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