  The ALEA Coq library formalizes measure theory based on a variant of the Giry
monad on the category of sets. This enables the interpretation of a
probabilistic programming language with primitives for sampling from discrete
distributions. However, continuous distributions have to be discretized because
the corresponding measures cannot be defined on all subsets of their carriers.
  This paper proposes the use of synthetic topology to model continuous
distributions for probabilistic computations in type theory. We study the
initial $\sigma$-frame and the corresponding induced topology on arbitrary
sets. Based on these intrinsic topologies we define valuations and lower
integrals on sets, and prove versions of the Riesz and Fubini theorems. We then
show how the Lebesgue valuation, and hence continuous distributions, can be
constructed.
