My Math Forum  

Go Back   My Math Forum > College Math Forum > Real Analysis

Real Analysis Real Analysis Math Forum


Reply
 
LinkBack Thread Tools Display Modes
September 11th, 2014, 11:07 AM   #1
Member
 
Joined: Mar 2013

Posts: 71
Thanks: 4

problem about measure

Please, check my solution to this problem:

"Prove that if $X \subset [a,b]$ isn't a measure-zero set, then there exists $\varepsilon >0$ such that, for every partition $P$ of $[a,b]$, the sum of the lenghts of the intervals of $P$ which contain points of $X$ is greater than $\varepsilon$".

Solution:

Let $X \subset [a,b]$ be a non measure-zero set. Then there exists $\varepsilon>0$ and an enumerable family of open intervals $I_1,...,I_n,...$ such that $X \subset I_1 \cup ... \cup I_n \cup ...$ with $\sum_{i=1}^{\infty} |I_i|\ge \varepsilon$.

Without lost of generality, We can supose $I_1 \cup ... \cup I_n \cup ... \subset [a,b]$. In such a case, the points $a$, $b$ and the extremities of the intervals $I_1,...,I_n,...$ form a partition of $[a,b]$. By construction, the intervals of this partition which contain points of $x$ have length $ \ge \varepsilon$.
Thank you for your attention!
walter r is offline  
 
Reply

  My Math Forum > College Math Forum > Real Analysis

Tags
measure, problem



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
measure DuncanThaw Real Analysis 4 November 23rd, 2013 11:02 AM
Measure limes5 Real Analysis 4 July 26th, 2013 12:47 PM
Lebesgue measure theory problem Jeh Real Analysis 0 July 19th, 2012 09:13 PM
Measure Problem Halmos everk Real Analysis 1 October 18th, 2010 01:39 PM
measure zero eskimo343 Real Analysis 1 December 6th, 2009 07:44 PM





Copyright © 2018 My Math Forum. All rights reserved.