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

[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 ?

Homework Equations

The Attempt at a Solution

 

[ystael]

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.

 

[sbashrawi]

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

 

[arkajad]

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

 

[sbashrawi]

I mean lebesgue measure

 

[ystael]

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.

 

[arkajad]

I mean lebesgue measure

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