Measure Theory/Integrable Almost Continuous

Integrable Is Almost Continuous

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In this lesson we will prove that if   is integrable then for every   there is a continuous function   such that

 

Integrable Is Approximately Simple

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Going directly toward that goal doesn't have an immediately clear path. Instead, we set a simpler goal, literally. We start by showing the corresponding result for simple functions, rather than continuous functions.

That is to say, here we will show that under the same conditions, there is a simple   such that

 

Because we assume f is integrable, as a general measurable function, we need to consider two integrals,   and  . We know that each of these is finite and nonnegative.

By their definitions as nonnegative integrals, there exist simple functions such that   and   and

 


Exercise 1. Approximate Simple Integration


Finish the proof started above. In particular, show that there is a simple function  , such that   is arbitrarily small.

(I am being deliberately coy about what the function   is. It must be somehow related to   but I want you to think about exactly how.)

Integrable is Approximately Step

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Exercise 3. Approximate Step Integration is the focus of this subsection. However, we will need the result of Exercise 2. Restricting the Domain Makes Integrals Small will be helpful for us to complete Exercise 3.


Exercise 2. Restricting the Domain Makes Integrals Small


Let   be an integrable function and   arbitrary.

Show that there exists a   such that for every measurable subset   with   we have

 

Hint: Use   and the monotone convergence theorem to find a simple function   such that  .

Now select   small enough to make  .

Use these ingredients to split   into   and proceed with the properties of integrals.


Exercise 3. Approximate Step Integration


Use the earlier result relating simple to step functions, to show that there is a step function s such that

 

with f and   as before.

Hint: Split the integral into two domains, one of which has a small domain and the other has a small integrand. Show that both terms are small.

Integrable Is Approximately Continuous

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Exercise 4. Approximate Continuous Integration


With the assumptions as before, now show that there is a continuous function g such that   is small. Given what we've shown above, at some point in your solution you should observe that for any simple function  ,

 

and the left-hand term can be chosen to be small. After making this choice, you should be able to then make a choice of g which makes the second term small.