Why is the sum of two simple functions also a simple function?

In summary, The sum of two simple functions is also simple because it can be rewritten as a linear combination of indicator functions of measurable sets, which is the definition of a simple function. This can be easily proven by induction.
  • #1
seeker101
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  • #2
seeker101 said:
I can intuitively see why the sum of two http://en.wikipedia.org/wiki/Simple_function#Definition" is also simple. But can someone point me to a formal proof?

What is there to prove? The definition says a simple function is a linear combination of indicator functions of measurable sets. A sum of two of them is just a longer linear combination of indicator functions of measurable sets.
 
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  • #3
Sorry. I left something out.

Suppose now the definition of a http://en.wikipedia.org/wiki/Simple_function#Definition" also requires the events [tex]A_k[/tex] to be mutually exclusive.

How can we now show that the sum of two simple functions will also be simple?
 
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  • #4
seeker101 said:
Sorry. I left something out.

Suppose now the definition of a http://en.wikipedia.org/wiki/Simple_function#Definition" also requires the events [tex]A_k[/tex] to be mutually exclusive.

How can we now show that the sum of two simple functions will also be simple?

It looks equivalent to me to the following statement:

Suppose:

[tex] A = \cup_{i=1}^n A_i[/tex]

where the Ai are measurable sets. Then A can be rewritten a

[tex] A = \cup_{i=1}^n B_i[/tex]

where the Bi are disjoint measurable sets. And this looks trivial to prove by induction.
 
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1. How can the sum of two simple functions produce another simple function?

The sum of two simple functions can produce another simple function because simple functions are defined as having a finite number of operations and constants. When two simple functions are added together, the resulting function will also have a finite number of operations and constants, therefore it is also a simple function.

2. Can the sum of two simple functions ever be a complex function?

No, the sum of two simple functions can never result in a complex function. As mentioned before, simple functions are defined as having a finite number of operations and constants. Complex functions, on the other hand, have an infinite number of operations or constants, which is not possible to obtain from the sum of two simple functions.

3. Is there a limit to the number of simple functions that can be added together?

Yes, there is a limit to the number of simple functions that can be added together. This limit is determined by the maximum number of operations and constants that can be included in a simple function. Once this limit is reached, the resulting function will no longer be considered a simple function.

4. Can the sum of two simple functions produce a function that is not continuous?

No, the sum of two simple functions will always produce a function that is continuous. This is because simple functions are defined as having a finite number of operations and constants, which means there are no breaks or jumps in the function. Therefore, adding two simple functions together will not introduce any discontinuities.

5. Are there any exceptions to the rule that the sum of two simple functions is also a simple function?

Yes, there are some exceptions to this rule. If one or both of the simple functions being added together are not well-defined or have infinite values, the resulting function may not be considered a simple function. Additionally, if the two simple functions have different domains, the resulting function may not be considered a simple function. In these cases, the sum of two simple functions would not result in a simple function.

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