Invariant subspaces of $RL^1$

In this note we extend D. Singh and A. A. W. Mehannas invariant subspace theorem for $RH^1$ (the real Banach space of analytic functions in $H^1$ with real Taylor coefficients) to the simply invariant subspaces of $RL^1$ (the real Banach space of functions in $L^1$ with real Fourier coefficients).