Proving Measure Zero for Set A Derived from Real Numbers Set E

In summary, To prove that set A has measure zero, we can approximate E by open intervals and then square them to obtain an approximation of A. However, since intervals are not bounded, we need to split the set into countably many bounded sets of measure zero. This will allow us to find a bound and prove that A also has measure zero.
  • #1
arvindam
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)
 
Physics news on Phys.org
  • #2
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
But intervals are not bounded. b2i-a2i does not have a bound when b--ai is bounded.
 
  • #4
arvindam said:
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.
 
  • #5


To prove that set A has measure zero, we can use the definition of measure zero and the properties of sets derived from real numbers.

First, we know that a set E has measure zero if for any epsilon greater than 0, there exists a countable collection of intervals whose total length is less than epsilon, and the union of these intervals covers the set E.

Now, for set A={x2 : x\inE}, we can see that it is a subset of the set of squares of real numbers, which also has measure zero. This means that for any epsilon greater than 0, there exists a countable collection of intervals whose total length is less than epsilon, and the union of these intervals covers the set of squares of real numbers.

We can then apply this property to set A, since it is a subset of the set of squares of real numbers. This means that for any epsilon greater than 0, there exists a countable collection of intervals whose total length is less than epsilon, and the union of these intervals covers the set A.

Additionally, since E is an unbounded subset of R, we can use the fact that the set of squares of real numbers is also unbounded. This means that for any interval, we can find another interval containing it, such that the length of the second interval is greater than the square of the length of the first interval. This property ensures that the total length of the countable collection of intervals covering the set A will always be less than epsilon, regardless of the choice of epsilon.

Therefore, we can conclude that set A has measure zero, since it satisfies the definition of measure zero for any epsilon greater than 0. This proof utilizes the properties of sets derived from real numbers and the unboundedness of E to show that set A has measure zero.
 

1. What is "Proving Measure Zero for Set A Derived from Real Numbers Set E"?

"Proving Measure Zero for Set A Derived from Real Numbers Set E" is a mathematical concept that involves proving that a specific set, A, which is derived from a set of real numbers, E, has a measure of zero. This means that the set A has no volume or area, and therefore contains no points.

2. Why is proving measure zero important in mathematics?

Proving measure zero is important in mathematics because it helps us understand the size and structure of different sets. It allows us to identify and classify sets based on their measure, and provides a way to measure the complexity of mathematical objects.

3. What is the process for proving measure zero for a set derived from real numbers?

The process for proving measure zero for a set derived from real numbers involves showing that the set can be covered by a countable number of intervals or "boxes" with arbitrarily small widths. This means that the set can be broken down into smaller and smaller pieces, and each piece can be made arbitrarily small, ultimately leading to a measure of zero.

4. How is proving measure zero related to the concept of "null set"?

The concept of a null set, also known as an empty set, is closely related to proving measure zero. A null set is a set that contains no elements, and therefore has a measure of zero. Similarly, a set with a measure of zero is essentially empty, as it contains no points.

5. Can measure zero be proven for any set derived from real numbers?

No, measure zero can only be proven for certain types of sets derived from real numbers. These include sets with fractal or self-similar structures, such as the Cantor set, and sets with infinitely many "gaps" or holes, such as the set of rational numbers. Sets with smooth or continuous structures, such as intervals or circles, do not have a measure of zero.

Similar threads

Replies
85
Views
4K
Replies
0
Views
364
  • Set Theory, Logic, Probability, Statistics
Replies
15
Views
2K
  • Differential Geometry
Replies
3
Views
2K
  • Differential Geometry
Replies
9
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
7
Views
1K
Replies
8
Views
2K
  • General Math
Replies
3
Views
838
  • Topology and Analysis
Replies
2
Views
149
  • Calculus and Beyond Homework Help
Replies
20
Views
2K
Back
Top