Expand integration and undesirable points

In summary, the conversation is discussing the idea of expanding the integration space to include functions with undefined points, and whether or not this approach is more general than using the measurement set theory. The speaker is suggesting to keep the integration in the form of a combination of subsets, regardless of whether they are countable or not. However, the other participant is unclear about the speaker's approach and mentions that it is much weaker than Lebesque integration, which can integrate over all measurable sets. The conversation ends with the speaker giving an example and the other participant expressing confusion about the relevance of "undesirable points".
  • #1
husseinshimal
10
0
The purpose behinde this post is an attempt to expand integration space to include functions with undesirable points where the function could be undefined.I need help to understand if this post makes sense.The main question here, is this way could be more general than using the measurement set theory to expand the integration?

Assume the function F(x),

F:[a,b]→R
We will put the interval[a,b] as acombination of subsets, UGk,




UGk=GNUGQUGQ̀...etc.(U,here stands for combination symbol) which stands for natural set,rational set ,irrational set,….etc.




Lets define the subsets,

өi={gk:F(xi)≥gk≥0},

xiЄ[a,b],

Let,(pi) be apartitoning of Gk in [a,b] and (pөi) apartitioning of(өi),


(Mөi=SUPөi), and (mөi=infөi),

(UFөi,pi)= Σi Mөi(xi-xi-1), (UFөi,pi)=upper darboux sum.

(LFөi,pi)= Σi mөi(xi-xi-1),(LFөi,pi)=lowe darboux sum.

(UFөi,pi,Pөi)=inf{UFөi:piPөi,(Pөi) partioning of(өi), (pi) partitioning of (Gk)},

(LFөi,pi,Pөi)=sup{LFөi:piPөi,(Pөi) partioning of(өi),(pi) partitioning (Gk)},

now, we put the integration in the form,

∫Fөi,over,Gk={0,UFөi}={0,LFөi}=the subsets,sөi,


∫F,over,[a,b]=Usөi


for example;

f(x):[0,1]→R,

f(x)=x , xЄ irrational numbers=Q̀, within[0,1],

f(x)=1,xЄ rational numbers=Q, within[0,1],

∫F,over,[o,1]={0,1\2}Q̀,U{0,1}Q={0,1\2}R,U,{1\2,1}Q , ,i.e,the integration would involve the real numbers within {0,1\2} plus the rational numbers within {1\2,1},(Q̀,Q and,R,are suffixes her and, U, stands for combination symbol.)

one might say it seems to be similar to the difference between Riemann integration and Lebesgue integration.

in Lebesgue integration,the above example would be, μ(0,1\2)inQ`set+μ(0,1)inQset=1\2+0=1\2,iam suggesting to keep integration in form of combination of subsets regardless they were countable or not.wouldnot this be more genral?
 
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  • #2
Frankly, I'm not at all clear what you are doing. You seem to be staying with intervals with the exception of "undesirable points". If so, your integration is much weaker than Lebesque integration in which we can integrate over all measurable sets. Other than the fact that all countable sets have measure 0 and so are trivial in Lebesque integration, I don't see what "countable or not" has to do with it.

The Lebesque integral of your f (f(x)= x if x is irrational, 0 if rational, 0< x< 1) is trivially 1/2. I have no idea why you would worry about any "undesirable points".
 

Related to Expand integration and undesirable points

1. What is integration and why is it important?

Integration is the process of combining different systems or components to function as a unified whole. It is important because it allows for efficient and effective communication between different parts of a system, leading to improved performance and functionality.

2. What are the benefits of expanding integration?

Expanding integration can lead to increased efficiency, improved data accuracy, and enhanced decision-making capabilities. It can also help streamline processes and reduce costs.

3. How can integration help address undesirable points?

Integration can help identify and address undesirable points by providing a comprehensive view of the system and its components. This allows for quicker identification and resolution of issues, leading to a more stable and reliable system.

4. What are some challenges of expanding integration?

Some challenges of expanding integration include compatibility issues between different systems, data security concerns, and the need for specialized expertise to implement and maintain the integrated system.

5. How can we ensure the success of an expanded integration project?

To ensure the success of an expanded integration project, it is important to have a clear understanding of the goals and objectives, involve all stakeholders in the planning and implementation process, and regularly monitor and evaluate the integration to make necessary adjustments. Proper training and support for users is also crucial for the successful adoption of the integrated system.

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