Measure Theory/Integration

Let be a -finite measure space. Suppose is a positive simple measurable function, with ; are disjoint.

Define

Let be measurable, and let .

Define

Now let be any measurable function. We say that is integrable if and are integrable and if . Then, we write


The class of measurable functions on is denoted by

For , we define to be the collection of all measurable functions such that


A property is said to hold almost everywhere if the set of all points where the property does not hold has measure zero.

PropertiesEdit

Let   be a measure space and let   be measurable on  . Then

  1. If  , then  
  2. If  ,  , then  
  3. If   and   then  
  4. If  ,  , then  , even if  
  5. If  ,  , then  , even if  

Proof


Monotone Convergence TheoremEdit

Suppose   and   are measurable for all   such that

  1.   for every  
  2.   almost everywhere on  

Then,  


Proof


  is an increasing sequence in  , and hence,   (say). We know that   is measurable and that  . That is,

 

Hence,  


Let  

Define  ;  . Observe that   and  

Suppose  . If   then   implying that  . If  , then there exists   such that   and hence,  .

Thus,  , therefore  . As this is true if  , we have that  . Thus,  .

Fatou's LemmaEdit

Let   be measurable functions. Then,

 

Proof

For   define  . Observe that   are measurable and increasing for all  .

As  ,  . By monotone convergence theorem,

  and as  , we have the result.

Dominated convergence theoremEdit

Let   be a complex measure space. Let   be a sequence of complex measurable functions that converge pointwise to  ;  , with  

Suppose   for some   then

  and   as  

Proof

We know that   and hence  , that is,  

Therefore, by Fatou's lemma,  


 

As  ,   implying that  

TheoremEdit

  1. Suppose   is measurable,   with   such that  . Then   almost everywhere  
  2. Let   and let   for every  . Then,   almost everywhere on  
  3. Let   and   then there exists constant   such that   almost everywhere on  

Proof

  1. For each   define  . Observe that  
    but   Thus   for all  , by continuity,   almost everywhere on  
  2. Write  , where   are non-negative real measurable.
    Further as   are both non-negative, each of them is zero. Thus, by applying part I, we have that   vanish almost everywhere on  . We can similarly show that   vanish almost everywhere on  .