  Distribution function is essential in statistical inference, and connected
with samples to form a directed closed loop by the correspondence theorem in
measure theory and the Glivenko-Cantelli and Donsker properties. This
connection creates a paradigm for statistical inference. However, existing
distribution functions are defined in Euclidean spaces and no longer convenient
to use in rapidly evolving data objects of complex nature. It is imperative to
develop the concept of distribution function in a more general space to meet
emerging needs. Note that the linearity allows us to use hypercubes to define
the distribution function in a Euclidean space, but without the linearity in a
metric space, we must work with the metric to investigate the probability
measure. We introduce a class of metric distribution functions through the
metric between random objects and a fixed location in metric spaces. We
overcome this challenging step by proving the correspondence theorem and the
Glivenko-Cantelli theorem for metric distribution functions in metric spaces
that lie the foundation for conducting rational statistical inference for
metric space-valued data. Then, we develop homogeneity test and mutual
independence test for non-Euclidean random objects, and present comprehensive
empirical evidence to support the performance of our proposed methods.
