  We study the off-diagonal current fluctuation in a macroscopic quantum system
measured in an ideal manner that is as close as possible to the classical ideal
measurement. We show rigorously that not only extended but also localized
states contribute to the off-diagonal current fluctuation. This result
contrasts with the fact that only the extended states affect the off-diagonal
(Hall) conductivity and apparently contradicts the naive expectation from the
fluctuation-dissipation theorem that might directly connect these two
quantities. More specifically, we study the off-diagonal current fluctuation in
a disordered two-dimensional electron system in a strong magnetic field at low
temperatures. The fluctuation is almost unchanged from that of the pure system
reflecting the property mentioned above, being approximately proportional to
the Landau level filling factor with high accuracy. Our finding paves the way
to estimate the filling factor and the electron density from the off-diagonal
current fluctuation observed in macroscopic systems.
