We investigate a tunnel contact coupled to a double quantum dot (DQD) and employed as charge monitor for the latter. We consider both the classical limit and the quantum regime. In the classical case, we derive measurement correlations from conditional probabilities, which yields quantitative statements about the parameter regime in which the detection scheme works well. Moreover, we demonstrate that not only the DQD occupation but also the corresponding current may strongly correlate with the detector current. The quantum mechanical solution, obtained with a Bloch-Redfield master equation, shows that the backaction of the measurement tends to localize the DQD electrons and, thus, significantly reduces the DQD current. Moreover, it provides the effective parameters of the classical treatment. It turns out that already the classical description is adequate for most operating regimes.