The integrability of conjugate functions

For any function $f$ of $L(0,2\pi)$, we prove that there is a function $\varphi\in L(0,2\pi)$ such that $|\varphi(x)|=|f(x)|$ almost everywhere and $\tilde{\varphi}\in L(0,2\pi)$, where $\tilde{\varphi}$ is the conjugate of $\varphi$.