Let $g$ be the Fourier transform of an unknown function $y\in L_1(-\infty,\infty)$:$$g(\lambda)=\int_{\mathbb{R}}y(x)e^{-i\lambda x}d\mu_x$$Let $f$ be defined as $$f(x):=\frac{1}{2\pi}\lim_{N\to\infty}\int_{-N}^Ng(\lambda)e^{i\lambda x}d\lambda.$$I think that $f$ does not exist in general for any $y\in L_1(-\infty,\infty)$: am I right? The book that I am studying on, Kolmogorov-Fomin's, does not touch the issue of its existence in general, but I think it does not.
Nevertheless, under some conditions $f$ does exist and (ex.: under the Dini condition for $y$ in $x$) we can even have $f(x)=y(x)$. If $f(x)$ exist for all $x\in\mathbb{R}$, then is it true that $$g(\lambda)=\int_{\mathbb{R}}f(x)e^{-i\lambda x}d\mu_x$$in general? Thank you so much for any answer!