Integral over a set of measure 0

In summary: Sorry that I cannot get your idea exactly. Or can you tell me explicitly what is the function f to be integrated?
  • #1
Boot20
10
0
Is the integral over a set of measure zero always equals to zero? Can the integral be undefined?
 
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  • #2
It may be undefined if the function itself is peculiar with infinity as its value. For ordinary functions the integral will be 0.
 
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  • #3
mathman said:
It may be undefined if the function itself is peculiar with infinity as its value. For ordinary functions the integral will be 0.

Sorry, I have the question that, if Lebesgue integration, they always define the convention [tex] \infty \cdot 0 = 0 [/tex], so, in this case, even the function takes [tex] \infty [/tex] in a set of measure 0, the integral is still 0?
 
  • #4
Convention is an easy way out.
 
  • #5
For Lebesgue integration, that the integral over a set of measure zero is a rather trivial theorem, following from the fact that all simple functions have integral zero -- so via (what I believe is) the usual formulation, it doesn't even need to be treated as a special case.


Riemann integration assumes the function is real-valued, so it doesn't even apply if you are considering extended-real-number-valued functions that take on the values [itex]+\infty[/itex] or [itex]-\infty[/itex].
 
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  • #6
But apparently to me [tex] \infty \cdot 0 = 0 [/tex] should be adopted
Else, if [tex] f [/tex] admit [tex] \infty [/tex] on set [tex] A [/tex] of measure [tex] 0 [/tex], we may use [tex] f_{n} = n [/tex] on [tex] A [/tex] to approximate [tex] f [/tex] from below, then, the integral of [tex] f_{n} [/tex] is zero, by monotone convergence theorem, the integral of [tex] f [/tex] should be zero as well. If we do not define [tex] \infty \cdot 0 = 0 [/tex], we may get inconsistency in this case?
 
  • #7
wayneckm said:
But apparently to me [tex] \infty \cdot 0 = 0 [/tex] should be adopted
Else, if [tex] f [/tex] admit [tex] \infty [/tex] on set [tex] A [/tex] of measure [tex] 0 [/tex], we may use [tex] f_{n} = n [/tex] on [tex] A [/tex] to approximate [tex] f [/tex] from below, then, the integral of [tex] f_{n} [/tex] is zero, by monotone convergence theorem, the integral of [tex] f [/tex] should be zero as well. If we do not define [tex] \infty \cdot 0 = 0 [/tex], we may get inconsistency in this case?

But what if the f_{n} are defined over sets of measure non-zero, but that the sum of the measure of those sets converges to zero?
 
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  • #8
Boot20 said:
But what if the f_{n} are defined over sets of measure noe intgen-zero, but that the sum of the measure of those sets converges to zero?

Sorry that I cannot get your idea exactly. Or can you tell me explicitly what is the function [tex] f [/tex] to be integrated?
 

Related to Integral over a set of measure 0

1. What does it mean to integrate over a set of measure 0?

Integrating over a set of measure 0 means that the set of values being integrated is so small that it has a measure of 0. This typically occurs when the set has a finite number of points or is a subset of a set with a measure of 0.

2. Why is integrating over a set of measure 0 important in mathematics?

Integrals over sets of measure 0 play a crucial role in mathematical analysis and measure theory. They allow us to define and evaluate integrals on more complicated sets, such as fractals, which cannot be described by traditional methods.

3. Can an integral over a set of measure 0 be non-zero?

No, the integral over a set of measure 0 must be equal to 0. This is because the integral is defined as the limit of a sum of products, and if the set has a measure of 0, then the sum of products will also be equal to 0.

4. How is the integral over a set of measure 0 different from a regular integral?

The main difference between the two is that the integral over a set of measure 0 is always equal to 0, while a regular integral can have a non-zero value. Additionally, the integral over a set of measure 0 is often used to extend the concept of integration to more complex sets.

5. In what real-world applications is integrating over a set of measure 0 used?

Integrating over sets of measure 0 has various applications in fields such as physics, economics, and engineering. For example, it can be used to calculate the area under a curve with discontinuities or to analyze the distribution of wealth in a population.

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