It is currently Sat Sep 22, 2018 6:47 am


All times are UTC [ DST ]




Post new topic Reply to topic  [ 1 post ] 
Author Message
PostPosted: Wed May 24, 2017 8:11 pm 

Joined: Thu Dec 10, 2015 1:58 pm
Posts: 59
Location: India
Problem: Let, $f$ be a real-valued Lebesgue measurable function on $\mathbb{R}^k$, prove that there exists Borel functions $g$ and $h$ such that $\mu_k(\{x \in \mathbb{R}^k: g(x) \neq h(x)\}) = 0$ and $g(x) \le f(x) \le h(x)$ a.e. $[\mu_k]$.

(where, $\mu_k$ is the Lebesgue measure on $\mathbb{R}^k$)

Note: This is problem 14 from pg-59 of Rudin's Real and Complex Analysis book. The problem as stated in book is incorrect. It requires us to find Borel functions $g$ and $h$ such that $\mu_k(\{x \in \mathbb{R}^k: g(x) \neq h(x)\}) = 0$ and $g(x) \le f(x) \le h(x)$ for all $x \in \mathbb{R}^k$. It's interesting to construct a counter-example to the above proposition/disprove it.


Top
Offline Profile  
Reply with quote  

Display posts from previous:  Sort by  
Post new topic Reply to topic  [ 1 post ] 

All times are UTC [ DST ]


Mathimatikoi Online

Users browsing this forum: No registered users and 1 guest


You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum
You cannot post attachments in this forum

Search for:
Jump to:  
cron
Powered by phpBB® Forum Software © phpBB Group Color scheme created with Colorize It.
Theme created StylerBB.net