We introduce the space of grid functions, a space of generalized functions of nonstandard analysis that provides a coherent generalization both of the space of distributions and of the space of Young measures. We will show that in the space of grid functions it is possible to formulate problems from many areas of functional analysis in a way that coherently generalizes the standard approaches. As an example, we discuss some applications of grid functions to the calculus of variations and to the nonlinear theory of distributions. Applications to nonlinear partial differential equations will be discussed in a subsequent paper.