Is Measureability Necessary for Proof of Pointwise Limit of Simple Functions?

In summary, the conversation discusses a proof that every measurable function can be written as the pointwise limit of a sequence of simple functions. It questions the necessity of the function's measurability for the proof and suggests that the statement could apply to all functions. The proof relies on the use of measurable sets and simple functions, making the measurability of the limit-function crucial. Without it, the proof would not hold.
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http://www.proofwiki.org/wiki/Measurable_Function_Pointwise_Limit_of_Simple_Functions
The following proof is shown in my book too. Basically it states that every measureable function can be written as the pointwise limit of a sequence of simple functions. Now, my problem is I don't see where the measureability of the limit-function is used crucially - so is the measureability really strictly necessary for the proof and if so where is it used? For me the statement might as well be all functions can be written as the pointwise limit of a sequence of simple functions?
 
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  • #2
By Characterization of Measurable Functions and Sigma-Algebra Closed under Intersection, it follows that:
Ann2n={f≥n}
Ank={f≥k2−n}∩{f<(k+1)2−n}
are all Σ-measurable sets.
Hence, by definition, all fn are Σ-simple functions.

If f was not measurable, then you wouldn't know that the fns were actually simple functions
 
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FAQ: Is Measureability Necessary for Proof of Pointwise Limit of Simple Functions?

What is the definition of a limit of a simple function?

The limit of a simple function, also known as the limit of a sequence, is the value that a function or sequence approaches as its input approaches a specific value or "approach point". It is a fundamental concept in mathematics that is used to describe the behavior of a function near a particular input value.

How is the limit of a simple function calculated?

The limit of a simple function can be calculated by evaluating the function at different input values that are increasingly closer to the approach point. The limit is then the value that the function approaches as the input values get infinitely close to the approach point.

What is the significance of the limit of a simple function?

The limit of a simple function is significant because it allows us to understand the behavior of a function near a particular input value. It helps us determine if the function is continuous, has a specific value at the approach point, or is approaching infinity or negative infinity.

Can the limit of a simple function exist even if the function is not defined at the approach point?

Yes, the limit of a simple function can exist even if the function is not defined at the approach point. This is because the limit is not dependent on the actual value of the function at the approach point, but rather on the behavior of the function as the input values get closer to the approach point.

What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit only considers the behavior of the function as the input values approach the approach point from one direction, either from the left or the right. A two-sided limit, on the other hand, considers the behavior of the function as the input values approach from both directions, and the limit exists only if the one-sided limits from both directions are equal.

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