  We propose a method to construct the stochastic integral simultaneously under
a non-dominated family of probability measures. Path-by-path, and without
referring to a probability measure, we construct a sequence of
Lebesgue-Stieltjes integrals whose medial limit coincides with the usual
stochastic integral under essentially any probability measure such that the
integrator is a semimartingale. This method applies to any predictable
integrand.
