We present a deep I, Z photometric survey covering a total area of 1.12 deg2 of the σ Orionis cluster and reaching completeness magnitudes of I = 22 and Z = 21.5 mag. From I, I - Z color-magnitude diagrams we have selected 153 candidates that fit the previously known sequence of the cluster. They have magnitudes in the range I = 16-23 mag, which corresponds to a mass interval from 0.1 down to 0.008 M &sun; at the most probable age of σ Orionis (2-4 Myr). Using J-band photometry, we find that 124 of the 151 candidates within the completeness of the optical survey (82%) follow the previously known infrared photometric sequence of the cluster and are probably members. We have studied the spatial distribution of the very low mass stars and brown dwarf population of the cluster and found that there are objects located at distances greater than 30 arcmin to the north and west of σ Orionis that probably belong to different populations of the Orion's Belt. For the 102 bona fide σ Orionis cluster member candidates, we find that the radial surface density can be represented by a decreasing exponential function (sigma = sigma _{0} e^{-r/r_{0}}) with a central density of σ0 = 0.23 ± 0.03 objects arcmin-2 and a characteristic radius of r 0 = 9.5 ± 0.7 arcmin. From a statistical comparison with Monte Carlo simulations, we conclude that the spatial distribution of the objects located at the same distance from the center of the cluster is compatible with a Poissonian distribution and, hence, that very low mass stars and brown dwarfs are not mainly forming aggregations or sub-clustering. Using near-infrared JHK-band data from Two Micron All Sky Survey and UKIRT Deep Infrared Sky Survey and mid-infrared data from Infrared Array Camera/Spitzer, we find that about 5%-9% of the brown dwarf candidates in the σ Orionis cluster have K-band excesses and 30% ± 7% of them show mid-infrared excesses at wavelengths longer than 5.8 μm. These are probably related to the presence of disks, most of which are "transition disks." We have also calculated the initial mass spectrum (dN/dm) of σ Orionis from very low mass stars (~0.10 M &sun;) to the deuterium-burning mass limit (0.012-0.013 M &sun;), i.e., complete in the entire brown dwarf regime. This mass spectrum is a rising function toward lower masses and can be represented by a power-law distribution (dN/dm vprop m -α) with an exponent α of 0.7 ± 0.3 for an age of 3 Myr.