The signal recorded during the measurement of a mechanical system response is a mixture of a useful signal and random noises. On the one hand, these noises enter the investigated system directly, and on the other hand they are added to the signal during its transmission to the measuring device and its further processing. Thus, the measured signals are a stochastic function of the response and, consequently, provide only indirect, but the only information of the actual response of the investigated system. The problem is to determine most accurately the actual response of the investigated system by virtue of known outputs from the measuring system. As such a problem is not sufficiently determined in a general case, but it can be solved as an optimum estimate of actual response. It is coming to light, however, that in the meaning of the minimum mean square deviation of actual response and of this estimate the optimum a posteriori first stochastic moment of the response is conditioned by the measured signals. This article describes the principal characteristics of the mutually influenced investigated system and the measuring system. After a short analysis of the principal variant of the problem, assuming that all noises entering the system are Gaussian white noises, principal attention is afforded to the cases when the noises are described by more complicated autocorrelation functions. Cross correlation of noises, which are considered as additive only, is neglected.