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

• sbashrawi
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
sbashrawi

## 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 ?

## The Attempt at a Solution

This is true as far as it goes, but you need to say specifically why the sets $$f^{-1}(m, \infty)$$ and $$f^{-1}(-\infty, M)$$ are measurable sets.

They are measurable since they are inverses of borel sets ( intervals)

In you original post: Are you talking about Borel measurability or Lebesgue measurability?

I mean lebesgue measure

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.

sbashrawi said:
I mean lebesgue measure

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

## 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.

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