  We seek to define statistical solutions of hyperbolic systems of conservation
laws as time-parametrized probability measures on $p$-integrable functions. To
do so, we prove the equivalence between probability measures on $L^p$ spaces
and infinite families of \textit{correlation measures}. Each member of this
family, termed a \textit{correlation marginal}, is a Young measure on a
finite-dimensional tensor product domain and provides information about
multi-point correlations of the underlying integrable functions. We also prove
that any probability measure on a $L^p$ space is uniquely determined by certain
moments (correlation functions) of the equivalent correlation measure.
  We utilize this equivalence to define statistical solutions of
multi-dimensional conservation laws in terms of an infinite set of equations,
each evolving a moment of the correlation marginal. These evolution equations
can be interpreted as augmenting entropy measure-valued solutions, with
additional information about the evolution of all possible multi-point
correlation functions. Our concept of statistical solutions can accommodate
uncertain initial data as well as possibly non-atomic solutions even for atomic
initial data.
  For multi-dimensional scalar conservation laws we impose additional entropy
conditions and prove that the resulting \textit{entropy statistical solutions}
exist, are unique and are stable with respect to the $1$-Wasserstein metric on
probability measures on $L^1$.
