We have studied the electronic structure of epitaxially grown thin films of La$_{1-x}$Sr$_x$FeO$_3$ by {it in-situ} photoemission spectroscopy (PES) and x-ray absorption spectroscopy (XAS) measurements. The Fe 2$p$ and valence-band PES spectra and the O $1s$ XAS spectra of LaFeO$_3$ have been successfully reproduced by configuration-interaction cluster-model calculation and, except for the satellite structure, by band-structure calculation.From the shift of the binding energies of core levels, the chemical potential was found to be shifted downward as $x$ was increased. Among the three peaks in the valence-band spectra of La$_{1-x}$Sr$_x$FeO$_3$, the peak nearest to the Fermi level ($E_F$), due to the ``$e_{g}$ band, was found to move toward $E_F$ and became weaker as $x$ was increased, whereas the intensity of the peak just above $E_F$ in the O $1s$ XAS spectra increased with $x$. The gap or pseudogap at $E_F$ was seen for all values of $x$. These results indicate that changes in the spectral line shape around $E_F$ are dominated by spectral weight transfer from below to above $E_F$ across the gap and are therefore highly non-rigid-band-like.