Is There a Requirement for Integrability in This Proof for an Open Ball in R^n?

  • Thread starter Thread starter Oiler
  • Start date Start date
  • Tags Tags
    Function Proof
Click For Summary
SUMMARY

The discussion focuses on the integrability of the function f: B -> R, defined as f(x) = ||x||^-3 for the open unit ball A in R^n. It is established that f is integrable on B when n=2 due to its bounded nature within the interval (0, 1) and the use of an increasing union of compact sets. Conversely, for n=3, f is not integrable, highlighting the necessity of specific conditions for integrability that are not met in this case. The conversation emphasizes the importance of understanding the limits and definitions associated with integrability in different dimensions.

PREREQUISITES
  • Understanding of open sets and complements in R^n
  • Familiarity with the concept of integrability in mathematical analysis
  • Knowledge of compact sets and their properties
  • Basic understanding of polar and spherical coordinates
NEXT STEPS
  • Study the properties of integrable functions in R^n
  • Explore the implications of boundedness on integrability
  • Learn about the use of increasing unions of compact sets in analysis
  • Investigate the differences in integrability criteria between dimensions, particularly in R^2 and R^3
USEFUL FOR

Mathematics students, particularly those studying real analysis, and educators looking to deepen their understanding of integrability concepts in higher dimensions.

Oiler
Messages
2
Reaction score
0
A is the open unit ball in R^n. Let B be the compliment of A (R^n\A).
If f: B -> R is defined by f(x) = ||x||^-3... (where x is in B)

For n=2, using an increasing union of compact sets show that f is integrable on B.

For n=3, show that f is not integrable.
____________________________________________

Does an increasing union of sets here mean that each compact set must be contained entirely in the next? It seems clear here that f will be bounded for n=2 (from 0 to 1), and thus would suggest that it is integrable, but then why not n=3? I seem to be missing a requirement for f being integrable here, any help would be appreciated.
 
Physics news on Phys.org
I did this question (I'm assuming that you've just taken this from the assignment) by choosing an obvious increasing union of compact sets. Polar coordinates, spherical coordinates hmm...

The fact that f is bounded doesn't really tell you that much. If you look at the definition you need a certain limit to exist.
 

Similar threads

Prove that this series converges uniformly on R
  • · Replies 4 ·
Replies
4
Views
2K
Calculate this Integral around the Circular Path using Green's Theorem
  • · Replies 3 ·
Replies
3
Views
2K
Prove every subset of countable set is either finite or else countable
  • · Replies 1 ·
Replies
1
Views
2K
Prove ##1 + a=s(a)=a+1## for ##a \in \mathbb{N}##
  • · Replies 1 ·
Replies
1
Views
2K
Integration over a set in n-dimensional space
  • · Replies 1 ·
Replies
1
Views
2K
Prove ##a \cdot 1 = a = 1 \cdot a## for ##a \in \mathbb{N}##
  • · Replies 1 ·
Replies
1
Views
2K
Proof given ##x < y < z## and a twice differentiable function
  • · Replies 8 ·
Replies
8
Views
2K
Prove ##|\{ q\in\mathbb{Q}: q>0 \} |=|\mathbb{N}|##.
  • · Replies 3 ·
Replies
3
Views
2K
Prove ##(a+b)\cdot c=a\cdot c+b\cdot c## using Peano postulates
  • · Replies 3 ·
Replies
3
Views
2K
Find f s. t. ||f||=1 and f(x) < 1 with ||x||=1
  • · Replies 20 ·
Replies
20
Views
3K