Measure Theory/Approximations of Measurable Functions

Simple, Step, and Continuous Functions

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Exercise 1. Simple Functions are Approximately Step

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Let   be a simple function on the bounded interval [a,b]. Also let  .

Prove that there exists a step function s and measurable set F such that

s and   are equal on F, and F is nearly all of [a,b].

More formally, for all  , we have  .

And  .

Hint: Approximate each of  .

Exercise 2. Step Functions are Approximately Continuous

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Let s be any step function on a closed, bounded interval,   and  . Also let  .

Prove that there is a continuous function   and measurable set  , such that

 

Hint: There is a finite number of discontinuities of s.

Put a small enough neighborhood around each discontinuity. Outside of these neighborhoods, make f and s equal.

Inside of these neighborhoods, interpolate a linear function from one end to the other.

Approximations of Measurable Functions

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Here we study two kinds of approximation results. One defines a notion in which two functions are "basically the same" (at least for the purposes of integration). This is the idea of functions being equal "almost everywhere". For functions equal almost everywhere, one may replace one function by the other and the value of the integral is unchanged.

The other kind of approximation result is that every measurable function is (1) approximately continuous, and (2) approximately step. This means that measurable functions can be replaced by continuous or step functions, and although this changes the value of the integral, it does so "not too much".

Definition: almost everywhere, a.e.

Let   be two measurable functions. We say that they are equal almost everywhere if the set of points at which they are not equal is a null set. That is to say,

 

We also abbreviate the statement that f equals g almost everywhere by writing  

Exercise 3. Almost Equal to 0

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Show that the constant function 0, and the Dirichlet function   are equal almost everywhere.

A.e. Preserves Measurability

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Suppose   are two functions, and f is measurable, and suppose   We will prove that therefore g is measurable.

Let   and consider the sets

 
 
 

Exercise 4. A.e. Preserves Measurability

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Justify why the following sets are measurable, in this order:

1. F
2. H
3.  
4.  
5. G

Simple Functions Are Approximately Step

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In this section we will show that, if   is a simple function then there is a step function   and a set   such that   on E and  .

So let   be a simple function and let  .

As we proved in a previous lesson, each   is approximately an open set. That is to say, there is an open set   such that  .

Measurable Functions Are Approximately Continuous and Step

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We will now show that for any measurable function   and  , there exists a continuous function   such that

  on a set  

and  . When   is "very small" then E is almost the whole interval [a,b], and g is very close to f.

We will also prove that there exists a step function   such that

  on a set  

and  .


First we prove that there is an M such that   except on a set of measure less than  .

To do so, consider the sequence of sets

 

Exercise 5. Measurable Approximately Bounded

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First show that   is measurable for each  .

Next show that  .

Then use the continuity of measure to show that there is some M for which  . Define  .

Infer that this is the desired M.


Next we will show that there is a simple function   such that   except on F.

To do so, set   such that   and define the sets

 

and then define the function

 

Exercise 6. Simple Approximation Confirmation

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Show that   is simple and, except on F,  .


Finally, use all of the above (with the help of results proved in earlier exercises) to prove the desired result.