Let $f:\mathbb{R}\to \mathbb{C}$ be a Lebesgue measurable function. For each $z\in \mathbb{C}$ let $Arg (z)$ be the principal argument of $z$ (define it to be $0$ if $z=0$). Define $g(x)= \sqrt{|f(x)|}exp(\frac{1}{2} i Arg (f(x))$. Then $g(x)$ is measurable and $(g(x))^2=f(x)$.
I am sure the above statement is correct, but just wanted to confirm with you all. The only part of concern is, of course, whether the function $exp(\frac{1}{2} i Arg (f(x))$ is measurable.