  A stochastic model with a continuum of economic agents often involves shocks
at both macro and micro levels. This can be formalized by a continuum of random
variables that are conditionally independent given the macro level shocks.
Based on the framework of a Fubini extension, the results on the exact law of
large numbers and its converse for a con- tinuum of independent random
variables in [14] are extended to the setting with conditional independence
given general macro states. As an illustrative application, it is shown that
any ex ante efficient allocation in an asymmetric information economy with
general aggre- gate uncertainty has a (utility) equivalent allocation that is
incentive compatible, which generalizes the corresponding results in [15] to
the case with infinitely many states.
