Measure Theory/Properties of Nonnegative Integrals

Properties of Nonnegative Integrals

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Throughout this lesson you may assume that

  •  
  •   are measurable, nonnegative functions
  •  

Recall the definition of a nonnegative integral,

  where the supremum is taken over all functions   which are bounded and  .

Several of the basic properties of nonnegative integrals are familiar from the lesson on bounded integrals. Consistency, linearity, order-preserving.

Notice that it makes no sense to talk about theorems which have to do with absolute values, such as the triangle inequality, since the function and therefore the integral are nonnegative, and hence equal to their own absolute value.

We then prove what might debatably be the most essential theorem in all of measure theory, the monotone convergence theorem. It will be helpful to first prove Fatou's lemma.

It will turn out that in later studies, Fatou's lemma takes on a life of its own, so to speak.

Consistency

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We yet again have conflicting definitions of the integral. This time, if   and   is bounded and measurable and nonnegative, then   could mean either the bounded integral or the nonnegative integral. In this subsection, we will understand that   refers to the nonnegative integral, and therefore we use the notation   to refer to the bounded integral.

Exercise 1. Bounded-Nonnegative Consistency

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Prove that   for any function f as described above. The style of proof should be very similar to that used for Prove Simple-Bounded Consistency.

Linearity and Order Preserving

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Exercise 1. Prove Linearity and Order Preserving

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Prove linearity and order preserving. The proof should be very similar in style to the one given for bounded integrals.

Fatou's Lemma

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Let   be a sequence of nonnegative measurable functions and   on a set E, then Fatou's lemma states

 

It suffices to prove that   is an upper bound on the set of all   where   is bounded, and is nonzero only a finite-measure set.

For any such h, we will construct a new sequence of functions which runs somewhat "in parallel" with the sequence  .

 

so that therefore   and therefore   and therefore  .

Exercise 2. Why Liminf?

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In the last step above, we took the limit inferior of both sides of an inequality. Why did we not simply take the limit?

Hint: It has to do with the fact that the sequence of infima is a necessarily increasing sequence, for any "source" sequence. It also has to do with the fact that each term in this particular sequence is necessarily nonnegative.

Exercise 3. Prove Fatou's

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Prove Fatou's lemma by the following steps.

1. Prove that for any fixed   the limit  .

2. Apply the Bounded Convergence Theorem.

3. Use the fact that when a limit exists, it must equal its limit inferior.

Monotone Convergence Theorem

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The monotone convergence theorem has the same set of assumptions as Fatou's lemma, but then further imposes the assumption that the sequence is monotonically increasing:

  •   for each  

It then states that therefore the swaparoo follows:

 

Exercise 4. Prove the MCT

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Prove the MCT. An immediate and instinctive application of Fatou's lemma is a great first step.

Now use the fact that the sequence is increasing, to infer that

 

and then conclude the theorem.