  Direct-imaging observations of terrestrial exoplanets will enable their
atmospheric characterization and habitability assessment. Considering the
Earth, the key atmospheric signatures for the biosphere is O$_2$ and the
photochemical product O$_3$. However, this O$_2$-O$_3$ biosignature is not
detectable in the visible wavelengths for most of the time after the emergence
of oxygenic photosynthesis life (i.e., the Proterozoic Earth). Here we
demonstrate spectroscopic observations in the ultraviolet wavelengths for
detecting and characterizing O$_2$ and O$_3$ in Proterozoic Earth-like planets,
using ExoReL$^\Re$. For an O$_2$ mixing ratio 2 to 3 orders of magnitude less
than the present-day Earth, and an O$_3$ mixing ratio of $10^{-7}-10^{-6}$, we
find that O$_3$ can be detected and its mixing ratio can be measured precisely
(within $~1$ order of magnitude) in the ultraviolet ($0.25-0.4\ \mu$m) in
addition to visible-wavelength spectroscopy. With modest spectral resolution
($R=7$) and S/N ($\sim10$) in the ultraviolet, the O$_3$ detection is robust
against other potential gases absorbing in the ultraviolet (e.g., H$_2$S and
SO$_2$), as well as the short-wavelength cutoff between 0.2 and 0.25 $\mu$m.
While the O$_3$ detection does not rely on the near-infrared spectra, extending
the wavelength coverage to the near-infrared ($1-1.8\ \mu$m) would provide
essential information to interpret the O$_3$ biosignature, including the mixing
ratio of H$_2$O, the cloud pressure, as well as the determination of the
dominant gas of the atmosphere. The ultraviolet and near-infrared capabilities
should thus be evaluated as critical components for future missions aiming at
imaging and characterizing terrestrial exoplanets, such as the Habitable Worlds
Observatory.
