Measure Theory/Derivatives of Integrals

Lebesgue's Differentiation Theorem

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We return to that problem which sent us along this sequence of thoughts, which is the proof that if f is integrable then

 

almost everywhere on  .

This is sometimes referred to as "Lebesgue's Differentiation Theorem".

To begin the proof we let   be the set of points at which this equality fails. Therefore we intend to show that its measure is zero.

Let   and we will attempt to approximate A in such a way that we are able to show   is small.

Let   be any continuous function with  .

Let   and by definition  .

Not only is this limit not zero, but because the integrand is nonnegative, then the limit must be positive.

Moreover, we may consider the limsup, since this is guaranteed to exist and be positive. This limsup may exist only in the sense of being infinity, but if we still regard that as a kind of existence then we will still have that it exists in this way.


Exercise 1. Limsup Is Equivalent


Let   be any nonnegative function  . Prove that   if and only if  .

Bounding the Limsup

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For the chosen   there exists some   such that

 

But then

 


Exercise 2. Complete the Inequalities


Explain why  . Hint: Recall what we already discussed in a previous lesson about continuous functions.

Explain why  .

Then use the above to infer that  .


Exercise 3. Find Bounds on the Domain Subsets


Use this result above to conclude that either   or  .

Apply Markov's inequality to the set

 

and apply Hardy-Littlewood to

 

to infer that

 

for some appropriate constant C.

Then prove that the left-hand side is small. (Recall  .)

Finally, prove the final result using the continuity of measure. (You will need to replace   by a countable sequence like  .)

Derivative of the Integral

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We finally put a bow on the entire section, by proving the following. We may call it the fundamental theorem of calculus, for length-measure integration.

Theorem: Let   be an integrable function and let   be its area function. Then   a.e.


Exercise 4. Prove the Derivative of the Integral Theorem


Prove the theorem above. It should be as direct as: Set up the limit of the difference quotient, and use the above to evaluate the limit.