I'm trying to define the $L^p$ spaces in the unit circle(denoted as $\mathbb T$), as Rudin's Real and Complex analysis does in page 88. I've defined a measure in $\mathbb T$ via Riesz's representation theorem, with the functional $\Lambda f=(1/{2\pi})\int_{-\pi}^{\pi} f(e^{it})dt$ (Lebesgue measure) and I'm trying to prove that there's a correspondence between $2\pi$-periodic functions in $\mathbb{R}$ and functions in $\mathbb T$ that preserves measurability.
I've already proved that if $F\colon\mathbb T\rightarrow \mathbb{C}$ is measurable , then the function $f(t)=F(e^{it})$ is periodic and measurable (Lebesgue). But the converse seems a bit tricky, because the inverse (as a function) of $e^{it}$ is not continuous , and that is the important part of the identification.
Sorry for my bad english and thanks in advance!