The VLBI Space Observatory Programme (VSOP) mission is a Japanese-led project to study radio sources with submilliarcsecond angular resolution, using an orbiting 8 m telescope on board the satellite HALCA with a global Earth-based array of telescopes. A major program is the 5 GHz VSOP Survey Program, which we supplement here with Very Long Baseline Array observations to produce a complete and flux density-limited sample. Using statistical methods of analysis of the observed visibility amplitude versus projected (u, v) spacing, we have determined the angular size and brightness temperature distribution of bright radio emission from active galactic nuclei. On average, the cores have a diameter (full width, half-power) of 0.20 mas, which contains about 20% of the total source emission, and 14% ± 6% of the cores are less than 0.04 mas in size. About 20% ± 5% of the radio cores have a source frame brightness temperature Tb > 1.0 × 1013 K, and 3% ± 2% have Tb > 1.0 × 1014 K. A model of the high brightness temperature tail suggests that the radio cores have brightness temperatures ≈1 × 1012 K and are beamed toward the observer with an average bulk motion of β = 0.993 ± 0.004.