MHB 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
The discussion centers on proving that the pre-image of any Borel set under a measurable function f on a measure space (X, Λ, μ) is also measurable, meaning it belongs to the σ-algebra Λ. Participants are working through the problem and considering various interpretations and approaches to the proof. A solution will be provided later, as the community is still deliberating on the best method to demonstrate this property. The importance of understanding the relationship between measurable functions and Borel sets in measure theory is emphasized. The thread highlights the collaborative effort to reach a consensus on the proof.
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

MHB What is the maximal function of a finite Borel measure on $\Bbb R^n$?
  • · Replies 1 ·
Replies
1
Views
3K
MHB Is $\mathcal{M}(X)$ a Banach space?
  • · Replies 1 ·
Replies
1
Views
3K
MHB How Do You Solve This Integral for Positive Integer n?
  • · Replies 1 ·
Replies
1
Views
5K
MHB How can $L^1(\Bbb R)$ be isomorphic to an ideal in $M(\Bbb R)$?
  • · Replies 2 ·
Replies
2
Views
3K
MHB Is the Real Projective Plane's Gaussian Curvature Always Positive?
  • · Replies 1 ·
Replies
1
Views
3K
MHB How to Prove $X$ Has Lebesgue Measure Zero?
  • · Replies 1 ·
Replies
1
Views
3K
MHB What is the solution to this week's POTW?
  • · Replies 1 ·
Replies
1
Views
3K
MHB Can Even Maps Between n-Spheres Have Odd Homological Degrees?
  • · Replies 1 ·
Replies
1
Views
3K
MHB Is the Tensor Algebra of $\Bbb Z/n\Bbb Z$ Isomorphic to $\Bbb Z[x]/(nx)$?
  • · Replies 1 ·
Replies
1
Views
3K
MHB Is the mapping from $\mathcal{L}^p(X,\mu)$ to $\mathcal{L}^1(X,\mu)$ continuous?
  • · Replies 1 ·
Replies
1
Views
2K