  We present a study of discrete X-ray sources in and around the
bulge-dominated, massive Sa galaxy, Sombrero (M104), based on new and archival
Chandra observations with a total exposure of ~200 ks. With a detection limit
of L_X = 1E37 erg/s and a field of view covering a galactocentric radius of ~30
kpc (11.5 arcminute), 383 sources are detected. Cross-correlation with Spitler
et al.'s catalogue of Sombrero globular clusters (GCs) identified from HST/ACS
observations reveals 41 X-rays sources in GCs, presumably low-mass X-ray
binaries (LMXBs). We quantify the differential luminosity functions (LFs) for
both the detected GC and field LMXBs, whose power-low indices (~1.1 for the
GC-LF and ~1.6 for field-LF) are consistent with previous studies for
elliptical galaxies. With precise sky positions of the GCs without a detected
X-ray source, we further quantify, through a fluctuation analysis, the GC LF at
fainter luminosities down to 1E35 erg/s. The derived index rules out a
faint-end slope flatter than 1.1 at a 2 sigma significance, contrary to recent
findings in several elliptical galaxies and the bulge of M31. On the other
hand, the 2-6 keV unresolved emission places a tight constraint on the field
LF, implying a flattened index of ~1.0 below 1E37 erg/s. We also detect 101
sources in the halo of Sombrero. The presence of these sources cannot be
interpreted as galactic LMXBs whose spatial distribution empirically follows
the starlight. Their number is also higher than the expected number of cosmic
AGNs (52+/-11 [1 sigma]) whose surface density is constrained by deep X-ray
surveys. We suggest that either the cosmic X-ray background is unusually high
in the direction of Sombrero, or a distinct population of X-ray sources is
present in the halo of Sombrero.
