  A transfunction is a function which maps between sets of finite measures on
measurable spaces.
  Push-forward operators form one important class of examples of transfunctions
and are identified with their respective measurable functions. In this regard,
transfunctions are a generalization of measurable functions between measurable
spaces. Additionally, there are naturally arising transfunctions with nice
properties which are not measurable functions.
  Transfunctions which are weakly $\sigma$-additive (commutable with addition
over countable sequences of orthogonal measures) between second-countable
metric spaces are of particular interest and are primarily developed in this
paper. We study such transfunctions which are localized: sending source
measures carried by small open sets to target measures also carried by small
open sets.
  With the right settings and assumptions, we develop some theorems which
characterize continuous functions and measurable functions, and show that the
behavior of localized transfunctions can be approximated by measurable
functions and by continuous functions, but only up to some error. We also
characterize transfunctions that correspond to Markov operators.
  In our investigation of transfunctions we are motivated by several potential
applications, including Monge-Kantorovich transportation problem or population
dynamics that will be presented in some detail in this paper.
