Summary: If \((X^ N, Y^ N)\) on a probability space \((\Omega^ N, {\mathcal F}^ N, P^ N)\) converge in distribution to \((X,Y)\) on \((\Omega, {\mathcal F}, P)\), it is not necessarily true that the conditional expectations \(E^{P^ N} \{F(X^ N)\mid Y^ N\}\) converge in distribution to \(E^ P \{F(X)\mid Y\}\), even for bounded, continuous functions \(F\). The limits of the conditional expectations can be determined if it is possible to make an absolutely continuous change of probability measure, from \(P^ N\) to \(Q^ N\), so that, under \(Q^ N\), \(X^ N\) and \(Y^ N\) are independent. The aim of this paper is to extend this technique to the case where \(P^ N\) is ``non quite'' absolutely continuous with respect to this probability measure \(Q^ N\); but where there is a subset \(\Omega^ N_ 1\) of \(\Omega^ N\) such that \(P^ N\) restricted to \(\Omega^ N_ 1\) is absolutely continuous with respect to \(Q^ N\), and such that \(P^ N (\Omega^ N_ 1)\to 1\) as \(N\to\infty\). The convergence results for the conditional expectations are shown to be the same as when the absolutely continuous change of measure can be made directly -- the conditional expectations converge, but not necessarily to \(E^ P \{F(X)\mid Y\}\). A filtering application is examined, where a signal process \(X\) and observation \(Y\) satisfying a pair of stochastic differential equations, are approximated by the solutions \(X^ N\) and \(Y^ N\) of a pair of stochastic difference equations. If our approximating model employs noise increments with a density which must have compact support, conditions are given which ensure the expected convergence of the conditional expectations.