Measure Zero Sets: Proving \sigma(E) Has Measure Zero

In summary, the question asks if \sigma (E) has measure zero if E has measure zero. The poster is trying to prove that if \sigma (E) is not of measure zero, then there exists a point in E such that \sigma (\{e\}) has positive measure. They have already shown that if E is open or a G_{\delta} set, then \sigma (E) is also measurable. They are asking for help in solving this problem.
  • #1
Kindayr
161
0

Homework Statement


Let [itex]\sigma (E)=\{(x,y):x-y\in E\}[/itex] for any [itex]E\subseteq\mathbb{R}[/itex]. If [itex]E[/itex] has measure zero, then [itex]\sigma (E)[/itex] has measure zero.

The Attempt at a Solution


I'm trying to show that if [itex]\sigma (E)[/itex] is not of measure zero, then there exists a point in [itex]E[/itex] such that [itex]\sigma (\{e\})[/itex] that has positive measure. But i don't know if this actually proves the question.

I have already shown that if [itex]E[/itex] open or a [itex]G_{\delta}[/itex] set, then [itex]\sigma (E)[/itex] is also measurable. Can I use these to solve this?

Any help is appreciated.
 
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  • #2
The set you've defined, [itex]\sigma (E)[/itex] is a subset of R^2. [itex]\sigma (\{e\})[/itex] has zero measure. It's a line in R^2.
 

FAQ: Measure Zero Sets: Proving \sigma(E) Has Measure Zero

1. What is a measure zero set?

A measure zero set is a set that has a Lebesgue measure of 0, meaning that it has no size or volume in the traditional sense. It is also known as a null set or a set of measure zero.

2. Why is proving that a set has measure zero important?

Proving that a set has measure zero is important because it allows us to understand and analyze the properties of sets that have no size. This is useful in many areas of mathematics, particularly in measure theory and integration.

3. How is the measure of a set determined?

The measure of a set is typically determined using the Lebesgue measure, which is a mathematical concept that assigns a measure to subsets of Euclidean space. It is defined as the infimum of the sums of the volumes of covering intervals for a given set.

4. What are some techniques for proving that a set has measure zero?

Some common techniques for proving that a set has measure zero include using the definition of the Lebesgue measure, showing that the set can be covered by a countable number of smaller sets with measure zero, and using the properties of measurable functions and their preimages.

5. Are all sets with measure zero considered to be unimportant or trivial?

No, not all sets with measure zero are considered to be unimportant or trivial. In fact, many important and interesting mathematical objects, such as the Cantor set and the set of rational numbers, have measure zero. The concept of measure zero allows us to study and understand these objects in a rigorous and meaningful way.

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