Is f Measurable if E is a Measurable Set of Measure Zero?

So since E has measure zero, any subset of E must also have measure zero, and therefore f^{-1}(m, \infty) and f^{-1}(-\infty, M) must also have measure zero. Therefore, they are measurable sets.In summary, the conversation discusses the measurability of a bounded function on a measurable set of measure zero. The argument is made that since the sets f^{-1}(m, \infty) and f^{-1}(-\infty, M) are inverses of Borel sets, they are measurable. However, the need for a specific argument from the hypotheses is mentioned, and it is ultimately concluded that since every subset of a set of measure zero is
  • #1
sbashrawi
55
0

Homework Statement


If E is a measurable set of measure zero, and f is bounded function on E. Is f measurable?

I tried to prove this by saying that E = { x in E | m< f(x) <M}
= {x in E | f(x) > m }intersecting { x in E | f(x) < M } and these are measurable
so f is measurable. Am I right ?


Homework Equations





The Attempt at a Solution

 
Physics news on Phys.org
  • #2
This is true as far as it goes, but you need to say specifically why the sets [tex]f^{-1}(m, \infty)[/tex] and [tex]f^{-1}(-\infty, M)[/tex] are measurable sets.
 
  • #3
They are measurable since they are inverses of borel sets ( intervals)
 
  • #4
In you original post: Are you talking about Borel measurability or Lebesgue measurability?
 
  • #5
I mean lebesgue measure
 
  • #6
sbashrawi said:
They are measurable since they are inverses of borel sets ( intervals)

This is what you are asked to prove. You need to give a specific argument from the hypotheses why they are measurable sets.
 
  • #7
sbashrawi said:
I mean lebesgue measure

But the isn't every subset of of a set of measure zero measurable?
 

Related to Is f Measurable if E is a Measurable Set of Measure Zero?

1. What is the definition of measurability?

Measurability refers to the property of a set or function to have a well-defined measure or size. In other words, a measurable set or function can be assigned a numerical value that represents its size or extent.

2. What is a measurable set?

A measurable set is a set in a measure space that has a well-defined measure. In other words, the size or extent of the set can be determined using a measure function.

3. What does it mean for a set to have measure zero?

A set has measure zero if its size or extent is equal to zero. This means that the set contains no elements or has a negligible size compared to the measure space it is contained in.

4. Is it possible for a measurable set to have measure zero?

Yes, it is possible for a measurable set to have measure zero. This means that the set contains no elements or has a negligible size compared to the measure space it is contained in.

5. Is f measurable if E is a measurable set of measure zero?

It depends on the definition of f and the measure space it is defined in. If f is a function that maps elements from the measurable set E to a measurable set of measure zero, then f is also measurable. However, if f is a function that maps elements from E to a non-measurable set, then f may not be measurable.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
2K
  • Calculus and Beyond Homework Help
Replies
8
Views
253
  • Calculus and Beyond Homework Help
Replies
18
Views
1K
Replies
10
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
2K
  • Calculus and Beyond Homework Help
Replies
31
Views
2K
  • Topology and Analysis
Replies
1
Views
381
  • Calculus and Beyond Homework Help
Replies
12
Views
331
  • Calculus and Beyond Homework Help
Replies
2
Views
346
  • Calculus and Beyond Homework Help
Replies
2
Views
929
Back
Top