  For a separable finite diffuse measure space $\mathcal{M}$ and an orthonormal
basis $\{\varphi_n\}$ of $L^2(\mathcal{M})$ consisting of bounded functions
$\varphi_n\in L^\infty(\mathcal{M})$, we find a measurable subset
$E\subset\mathcal{M}$ of arbitrarily small complement $|\mathcal{M}\setminus
E|<\epsilon$, such that every measurable function $f\in L^1(\mathcal{M})$ has
an approximant $g\in L^1(\mathcal{M})$ with $g=f$ on $E$ and the Fourier series
of $g$ converges to $g$, and a few further properties. The subset $E$ is
universal in the sense that it does not depend on the function $f$ to be
approximated. Further in the paper this result is adapted to the case of
$\mathcal{M}=G/H$ being a homogeneous space of an infinite compact second
countable Hausdorff group. As a useful illustration the case of $n$-spheres
with spherical harmonics is discussed. The construction of the subset $E$ and
approximant $g$ is sketched briefly at the end of the paper.
