Let$ (X,M,\mu )$ is a measure space and $f,g$ are real valued functions on X .which of following options is true?
1) if $|f|$ be measurable then $f$ is measurable.
2) if $f^n$ (for all $n>4$ and $n \in \Bbb{N} $) be measurable then $f$ is measurable.
3)if $f+g$ be measurable then $f-g$ is measurable.
4) if $f×g$ be measurable and $g(x) \neq 0 $ then $ \frac{f}{g} $ is measurable.
I find counterexamples for 1 and 3 if we set $ f(x)= 1 $ if $ x \in A$ otherwise $f(x)=-1$ (that A is a non measurable subset of X) then $|f(x)|=1$ and 1 is false . For 3 $ f(x)= \chi _ A $ and $ g(x)= - \chi _ A $ and for 2 we know that $k(x)= \sqrt[5] {x} $ is measurable function so $f$ is measurable function .I can't find counterexample for 4.