  We introduce a new theory of generalised solutions which applies to fully
nonlinear PDE systems of any order and allows for merely measurable maps as
solutions. This approach bypasses the standard problems arising by the
application of Distributions to PDEs and is not based on either integration by
parts or on the maximum principle. Instead, our starting point builds on the
probabilistic representation of derivatives via limits of difference quotients
in the Young measures over a toric compactification of the space of jets. After
developing some basic theory, as a first application we consider the Dirichlet
problem and we prove existence-uniqueness-partial regularity of solutions to
fully nonlinear degenerate elliptic 2nd order systems and also existence of
solutions to the $\infty$-Laplace system of vectorial Calculus of Variations in
$L^\infty$.
