Measure Theory/Properties of General Integrals

Properties and Convergence of General Integrals

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Recall the definition of the length-measure integral of a measurable function  ,

 

and recall the definitions

 

In this lesson, assume

  •   are measurable functions
  •  
  •  
  •   for all  

Exercise 1. Consistency

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Explain why the check for consistency, which we have done for previous generalizations of integral definitions, is trivial in this case.

Exercise 2. Basic Properties of General Integrals

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Prove the basic properties of general integrals: Linearity, order-preserving, triangle inequality, the ML bound, and finite additivity.

Note: What is the positive part of   in terms of  ?

Lebesgue's Dominated Convergence Theorem

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Here resides the last (or, depending on how you count, the penultimate) of the great and famous convergence theorems of measure theory.

Definition: dominated sequence

Let   be a sequence of functions   and   another function. If for all   we have   then we say that g dominates the sequence  .

Besides the assumptions at the top of this page, further assume that g is integrable.

Lebesgue's Dominated Convergence Theorem then states that the swaparoo follows.

 

Exercise 3. Prove the LDCT

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 Part A. 
With the observation that   is a nonnegative function, use Fatou's lemma.
 Part B. 
Use an earlier result to establish that f is integrable.  Then infer from Part A. that  .
Part C.
Now apply reasoning similar to that in parts A. and B. above to   to infer   and conclude the proof.
Part D. 
Prove the following generalization of the LDCT.  

Let   be a sequence of integrable functions.  Assume further that 

*   so-to-speak "pairwise dominates" the sequence  . Formally this means   for each  .

*   on E.

*  

Now prove that  .

The proof should merely reiterate all of the proof of the LDCT, but replacing g by   where appropriate.