Measurable functions

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

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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

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.

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