  We introduce a new measure of ergodicity, the support set $S_\varepsilon$,
for random wave functions on disordered lattices. It is more sensitive than the
traditional inverse participation ratios and their moments in the cases where
the extended state is very sparse. We express the typical support set
$S_{\varepsilon}$ in terms of the distribution function of the wave function
amplitudes and illustrate the scaling of $S_{\varepsilon}\propto N^{\alpha}$
with $N$ (the lattice size) for the most general case of the multi-fractal
distribution. A number of relationships between the new exponent $\alpha$ and
the conventional spectrum of multi-fractal dimensions is established. These
relationships are tested by numerical study of statistics of wave functions on
disordered Bethe lattices. We also obtain numerically the finite-size spectrum
of fractal dimensions on the Bethe lattice which shows two apparent fixed
points as $N$ increases. The results allow us to conjecture that extended
states on the Bethe lattice at all strengths of disorder below the localization
transition are non-ergodic with a clear multifractal structure that evolves
towards almost ergodic behavior in the clean limit.
