  This work is the first in a series of papers devoted to the construction and
study of scaling limits of dynamical and near-critical planar percolation and
related objects like invasion percolation and the Minimal Spanning Tree. We
show here that the counting measure on the set of pivotal points of critical
site percolation on the triangular grid, normalized appropriately, has a
scaling limit, which is a function of the scaling limit of the percolation
configuration. We also show that this limit measure is conformally covariant,
with exponent 3/4. Similar results hold for the counting measure on macroscopic
open clusters (the area measure), and for the counting measure on interfaces
(length measure).
  Since the aforementioned processes are very much governed by pivotal sites,
the construction and properties of the "local time"-like pivotal measure are
key results in this project. Another application is that the existence of the
limit length measure on the interface is a key step towards constructing the
so-called natural time-parametrization of the SLE(6) curve.
  The proofs make extensive use of coupling arguments, based on the separation
of interfaces phenomenon. This is a very useful tool in planar statistical
physics, on which we included a self-contained Appendix. Simple corollaries of
our methods include ratio limit theorems for arm probabilities and the
rotational invariance of the two-point function.
