Function defined on a zero measured set

Question

I am trying to show that a function on a zero measured set is measurable My basic idea is that because i know every zero measured set is lebesgue measurable then i have to show every function with a measurable domain is measurable I dunno if i am right or wrong Please help me solve it ! Thanks

Answer 1

Hint: Every subset of a Lebesgue-measurable set of measure zero is Lebesgue-measurable (with measure zero).

Answer 2

Proposition (2) page 55 in Royden Real Analysis: Let the function f be defined on a measurable set E . Then f is measurable if and only if for each open set $O$ , the inverse image of $O$ under f , $f^{-1}$($O$)={$x \in $ E| $f (x) \in O$} is measurable . Now, since $f^{-1}$($O$) $\subseteq$ $E$ $\forall$ open set $O$ and m($E$)=0 ,but any subset of a set of measure zero is measurable so $f^{-1}$($O$) is measuable $\square$