  The Daniell-Kolmogorov Extension Theorem is a fundamental result in the
theory of stochastic processes, as it allows one to construct a stochastic
process with prescribed finite-dimensional distributions. However, it is
well-known that the domain of the constructed probability measure - the product
sigma-algebra in the set of all paths - is not sufficiently rich. This problem
is usually dealt with through a modification of the stochastic process,
essentially changing the sample paths so that they become c\`adl\`ag. Assuming
a countable state space, we provide an alternative version of the
Daniell-Kolmogorov Extension Theorem that does not suffer from this problem, in
that the domain is sufficiently rich and we do not need a subsequent
modification step: we assume a rather weak regularity condition on the
finite-dimensional distributions, and directly obtain a probability measure on
the product sigma-algebra in the set of all c\`adl\`ag paths.
