Measure Theory/Approximations of Sequences of Functions

Approximations of Sequences of Functions

edit

The content of this lesson is the final of Littlewood's three principles, which are also known as Egoroff's Theorem and Lusin's Theorem.

Pointwise Convergence to Nearly Uniform

edit

Here we will show that pointwise convergence can, through an appropriate domain restriction, entail something like uniform convergence.

There are a few theorems that show something like the rough expression above. The first interpretation that we will prove, in a slightly more formal expression, is:

For   as small as you like, you never need to remove more than   percent of the domain, for the convergence to be uniform.

The formal expression of this is: Let

  •   has finite measure,
  •   a sequence of measurable functions defined on E,
  •   pointwise on E,

Then   such that

  •  
  • on  
  •  

To begin the proof, let  .

With the assumptions as above, define for each   the set of points where the sequence is "far apart":

 

Exercise 1. Where the Sequence Is Far

edit

Definition: sup and limsup

Let   be any sequence of sets, each a subset of some set X. Define the supremum of   to be their union. Also define the limit superior of   to be the intersection over all suprema of tails:

 .

Of course every definition above for suprema has a correlate for infima.

Part A. Prove that the subsets of any set X are well-ordered by the subset relation.  
Part B. Prove that the union of   is an upper bound on this sequence, for the subset relation.  Also, and in the same sense, prove that it is the least of the upper bounds.
Part C. Explain why the limit supremum for real numbers is analogous to the limit supremum for sets.  There are several equivalent definitions of the limit supremum for a sequence of real numbers,  .  So take the definition to be 

 
Part D. State the definitions that make the most sense for infima and limit inferiors, for sets.  
Part E. Now prove that the limit superior of   is the empty set.
Part F. Define the union of the Nth tail,   and apply the continuity of measure, to infer that   such that  
Part G. Conclude the rest of the proof of the theorem.

Part H.

Definition: converge a.e.

Let   and   be a sequence of measurable functions defined on E. Let   be a measurable subset, such that   and   for all  . Then we say that   pointwise a.e.

Show that the result we finished proving in Part G. also holds if the condition of pointwise convergence is replaced by pointwise a.e. convergence.

Egoroff's Theorem

edit

Egoroff's theorem states the following.

Let   pointwise a.e. on a measurable set E of finite measure.

Then   such that

  •  
  •   on  .

Exercise 2. Prove Egoroff's

edit

Prove Egoroff's theorem.

Hint: Use the Part H. of Exercise 1. Where the Sequence Is Far together with the   trick.

Lusin's Theorem

edit

Lusin's theorem states the following.

Let

  •   be a measurable real-valued function.
  •  

Then there exists a continuous   such that  .

Exercise 3. Prove Lusin's

edit

Prove Lusin's theorem.

Hint: Use Egoroff's and the approximation of measurable functions by continuous functions.