Measure Theory/Properties of General Integrals
Properties and Convergence of General Integrals
[edit | edit source]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
[edit | edit source]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
[edit | edit source]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
[edit | edit source]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
[edit | edit source] 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.