Measurements of the turbulent magnetic field in a Hall thruster have been carried out between 1kHz and 30MHz with the aim of understanding electron transport through the magnetic field. Small detecting coils at the exit of the accelerating channel and outside of the ionic plume were used to characterize various instabilities. The characteristic frequencies of the observed power spectral densities correspond to known classes of instabilities: low frequency (20–40kHz), transit time (100–500kHz), and high frequency (5–10MHz). A model of the localized electron currents through a magnetic barrier is proposed for the high-frequency instability, and is found to be in good quantitative agreement with the observations. Based on the measured high-frequency turbulent magnetic field, the turbulent electric field is estimated to be about 1V∕cm outside of the plume and ranges from 10to102V/cm at the channel midradius at the exit of the thruster. The “anomalous” electron collision frequency, related to the high-frequency instability, is estimated to be <106s−1, which largely exceeds the classical frequency in the core of the exit plasma but is lower than the frequency that is generally used in hybrid codes