We report on the observation of incoherent Cherenkov radiation emitted by a 5.3 GeV positron beam circulating in the Cornell electron-positron storage ring as the beam passes in the close vicinity of the surface of a fused silica radiator (i.e., at a distance larger than 0.8 mm). The shape of the radiator was designed in order to send the Cherenkov photons towards the detector, consisting of a compact optical system equipped with an intensified camera. The optical system allows both the measurements of 2D images and angular distribution including polarization study. The corresponding light intensity has been measured as a function of the distance between the beam and the surface of the radiator and has shown a good agreement with theoretical predictions. For highly relativistic particles, a large amount of incoherent radiation is produced in a wide spectral range. A light yield of 0.8×10^{-3} photon per particle per turn has been measured at a wavelength of 600±10 nm in a 2 cm long radiator and for an impact parameter of 1 mm. This will find applications in accelerators as noninvasive beam diagnostics for both leptons and hadrons.