  The paper suggests a way of stochastic integration of random integrands with
respect to fractional Brownian motion with the Hurst parameter H> 1/2. The
integral is defined initially on the processes that are "piecewise" predictable
on a short horizon. Then the integral is extended on a wide class of square
integrable adapted random processes. This class is described via a mild
restriction on the growth rate of the conditional mean square error for the
forecast on an arbitrarily short horizon given current observations;
differentiability of H\"older property of any kind or degree is not required
for the integrand. The suggested integration can be interpreted as foresighted
integration for integrands featuring corresponding restrictions on the
forecasting error. This integration is based on It\^o's integration and does
not involve Malliavin calculus or Wick products. In addition, it is shown that
these stochastic integrals depend continuously on H at H=1/2+0.
