- #1
Boot20
- 10
- 0
Is the integral over a set of measure zero always equals to zero? Can the integral be undefined?
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.
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?
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?
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.
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.
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.
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.
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.