Do we have an $L^1$ function whose Fourier series converges almost everywhere but not to itself? 



An L1 function whose Fourier series converges but not to itself
The formulation seems specific enough to me. If the partial sums of the Fourier series converge to a function $g$ a.e., then so do the Cesaro means. But these converge in the $L^1$-sense to the original function. So the answer to your question is no.


