Set of Measure zero

  • Thread starter arvindam
  • Start date
  • #1
2
0
A set E subset of real numbers has measure zero. Set A={x2 : x[tex]\in[/tex]E}. How to prove that set A has measure zero?
(E could be any unbounded subset of R)
 

Answers and Replies

  • #2
22,089
3,296
I would approximate E by open intervals. That is: there exists open intervals [tex]]a_i,b_i[[/tex] such that

[tex]E\subseteq \bigcup]a_i,b_i[[/tex]

and such that

[tex]\sum{b_i-a_i}<\epsilon[/tex].

Now square the open intervals to obtain an approximation of A.
 
  • #3
2
0
But intervals are not bounded. b2i-a2i does not have a bound when b--ai is bounded.
 
  • #4
lavinia
Science Advisor
Gold Member
3,245
627
But intervals are not bounded. b2i-a2i does not have a bound when b--ai is bounded.

it is easy for any bounded set of measure zero. Split your set up into countably many of these.
 

Related Threads on Set of Measure zero

Replies
2
Views
2K
  • Last Post
Replies
5
Views
3K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
2
Views
3K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
8
Views
5K
  • Last Post
Replies
0
Views
3K
  • Last Post
Replies
1
Views
3K
Replies
4
Views
2K
  • Last Post
Replies
1
Views
2K
Top