Measurability of a function with finite codomain

  • Context: Graduate 
  • Thread starter Thread starter hwangii
  • Start date Start date
  • Tags Tags
    Finite Function
Click For Summary
SUMMARY

The discussion centers on the measurability of a function f defined from the interval X=[0,1] to a finite codomain Y, where |Y| < ∞. It is established that if f is the characteristic function of a non-measurable subset of the unit interval, then f is not Borel measurable. This conclusion highlights the importance of the nature of the subset in determining the measurability of functions in this context.

PREREQUISITES
  • Understanding of Borel measurability
  • Familiarity with characteristic functions
  • Knowledge of finite sets in mathematical analysis
  • Basic concepts of measure theory
NEXT STEPS
  • Study the properties of Borel sets and their applications
  • Explore the concept of non-measurable sets in measure theory
  • Learn about characteristic functions and their implications in analysis
  • Investigate the relationship between finite codomains and measurability
USEFUL FOR

Mathematicians, students of analysis, and researchers interested in measure theory and the properties of measurable functions.

hwangii
Messages
3
Reaction score
0
Hi all,
I have a simple question as follows:
[itex]f[/itex] is a function from [itex]X[/itex] to [itex]Y[/itex] where
[itex]X=[0,1][/itex]; and
[itex]Y[/itex] is finite, i.e. [itex]\vert Y\vert <\infty[/itex]
then is [itex]f[/itex] Borel measurable?
Thank you for your help in advance.
 
Physics news on Phys.org
hwangii said:
Hi all,
I have a simple question as follows:
[itex]f[/itex] is a function from [itex]X[/itex] to [itex]Y[/itex] where
[itex]X=[0,1][/itex]; and
[itex]Y[/itex] is finite, i.e. [itex]\vert Y\vert <\infty[/itex]
then is [itex]f[/itex] Borel measurable?
Thank you for your help in advance.
Why would you think that?
 
Let f be the characteristic function of a non-measurable subset of the unit interval. f is not Borel measurable.
 

Similar threads

Undergrad Simple partitioning of sequence in Proposition 1.15 in Folland's text
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Collection of finite unions of half-open intervals form an algebra
  • · Replies 3 ·
Replies
3
Views
4K
Undergrad Exercise 2.23 in Folland's real analysis text
  • · Replies 2 ·
Replies
2
Views
3K
MHB Prove that the function is monotonic and not decreasing
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Connection between absolute continuity of function and measure
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Are slices of measurable functions measurable?
  • · Replies 3 ·
Replies
3
Views
4K
Undergrad On commutativity of convolution
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Question about lifting of a function
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad On translation and dilation invariant Lebesgue measure: Folland's text
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad A concise "proof" of the Riemann-Lebesgue lemma
  • · Replies 1 ·
Replies
1
Views
3K