Simultaneous quantization of edge and bulk Hall conductivity

The edge Hall conductivity is shown to be an integer multiple of $e^2\mathrm{/}h$ which is almost surely independent of the choice of the disordered configuration. Its equality to the bulk Hall conductivity given by the Kubo-Chern formula follows from K-theoretic arguments. This leads to quantization of the Hall conductance for any redistribution of the current in the sample. It is argued that in experiments at most a few percent of the total current can be carried by edge states.