  We give a sufficient condition for the ergodicity of the Lebesgue measure for
an iterated function system of diffeomorphisms. This is done via the induced
iterated function system on the space of continuum (which is called
hyper-space). We introduce a notion of minimality for induced IFSs which
implies that the Lebesgue measure is ergodic for the original IFS. Here, to
beginning, the required regularity is $C^1$. However, it is proven that the
$C^1$-regularity is a redundant condition to prove ergodicity with respect to
the class of quasi-invariant measures.
  As a consequence of mentioned results, we obtain ergodicity with respect to
Lebesgue measure for several systems.
