Is the Pre-Image of a Measurable Function on a Measure Space also Measurable?

  • Thread starter Thread starter Chris L T521
  • Start date Start date
Click For Summary
SUMMARY

The discussion centers on proving that the pre-image of any Borel set in the real numbers, under a measurable function \( f \) defined on a measure space \( (X, \Lambda, \mu) \), is also a member of the sigma-algebra \( \Lambda \). Participants emphasize the importance of understanding the properties of measurable functions and Borel sets in this context. The solution is pending due to varying interpretations among contributors, indicating a need for clarity in definitions and proofs related to measurable functions.

PREREQUISITES
  • Understanding of measurable functions in measure theory
  • Familiarity with Borel sets and their properties
  • Knowledge of sigma-algebras and measure spaces
  • Basic concepts of real analysis
NEXT STEPS
  • Study the properties of measurable functions in detail
  • Explore the definition and construction of Borel sets
  • Learn about sigma-algebras and their role in measure theory
  • Review examples of pre-images of functions in measure spaces
USEFUL FOR

Mathematicians, students of real analysis, and anyone studying measure theory who seeks to deepen their understanding of measurable functions and their properties.

Chris L T521
Gold Member
MHB
Messages
913
Reaction score
0
Here's this week's problem!

-----

Problem: Let $f$ be a measurable function on a measure space $(X,\Lambda,\mu)$. Show that the pre-image of any Borel set of $\mathbb{R}$ is also in $\Lambda$.

-----

Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
Physics news on Phys.org
I will edit this post soon (by this afternoon PST) and update it with the solution then (since it's still being decided upon due to different interpretations).
 

Similar threads

Undergrad What is the maximal function of a finite Borel measure on $\Bbb R^n$?
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Is $\mathcal{M}(X)$ a Banach space?
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad How can $L^1(\Bbb R)$ be isomorphic to an ideal in $M(\Bbb R)$?
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad How Do You Solve This Integral for Positive Integer n?
  • · Replies 1 ·
Replies
1
Views
5K
High School What is the sum of the angles in a right triangle?
  • · Replies 1 ·
Replies
1
Views
2K
High School Is \(n^{n+1} > (n+1)^n\) for All \(n \geq 3\)?
  • · Replies 1 ·
Replies
1
Views
2K
High School What Is the Smallest Integer $a$ for $\frac{1}{640}=\frac{a}{10^b}$?
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad What is the solution to this week's POTW?
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad How to Prove $X$ Has Lebesgue Measure Zero?
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Is the Real Projective Plane's Gaussian Curvature Always Positive?
  • · Replies 1 ·
Replies
1
Views
3K