I am looking for a reference for the following result:
Let $f \in L^1(\Bbb R)$ be such that $\hat{f} \in L^1$. Then, $$f(x) = \int_{\Bbb R}\hat{f}(\lambda)e^{2\pi i\lambda x}\,{\mathrm d}\lambda$$ holds for all $x \in L_f$, where $L_f$ is the Lebesgue set of $f$ defined as $$\left\{x \in \Bbb R : \lim_{r \to 0}\frac{1}{r^2}\int_{B(x, r)}|f(y) - f(x)|\,{\mathrm d}x = 0\right\}.$$
In other words, I wish to show that the Fourier inversion formula holds for $x$ in the Lebesgue set of $f$. ($\hat f$ denotes the Fourier transform of $f$.)