  In this paper, we construct and then prove the up-to constants uniqueness of
the natural measure on several random fractals, namely the SLE cut points, SLE
boundary touching points, CLE pivotal points and the CLE carpet/gasket. As an
application, we also show the equivalence between our natural measures defined
in this paper (i.e. CLE pivotal and gasket measures) and their discrete analogs
of counting measures in critical continuum planar Bernoulli percolation in
[Garban-Pete-Schramm, J. Amer. Math. Soc.,2013]. Although the existence and
uniqueness for the natural measure for CLE carpet/gasket have already been
proved in [Miller-Schoug, arXiv:2201.01748], in this paper we provide with a
different argument via the coupling of CLE and LQG.
