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In the derivation of fluctuation relations, and in stochastic thermodynamicsin general, it is tacitly assumed that we can measure the system perfectly,i.e., without measurement errors. We here demonstrate for a driven systemimmersed in a single heat bath, for which the classic Jarzynski equality$\langle e^{-\beta(W-\Delta F)}\rangle = 1$ holds, how to relax thisassumption. Based on a general measurement model akin to Bayesian inference wederive a general expression for the fluctuation relation of the measured workand we study the case of an overdamped Brownian particle and of a two-levelsystem in particular. We then generalize our results further and incorporatefeedback in our description. We show and argue that, if measurement errors arefully taken into account by the agent who controls and observes the system, thestandard Jarzynski-Sagawa-Ueda relation should be formulated differently. Weagain explicitly demonstrate this for an overdamped Brownian particle and atwo-level system where the fluctuation relation of the measured work differssignificantly from the efficacy parameter introduced by Sagawa and Ueda.Instead, the generalized fluctuation relation under feedback control, $\langlee^{-\beta(W-\Delta F)-I}\rangle = 1$, holds only for a superobserver havingperfect access to both the system and detector degrees of freedom,independently of whether or not the detector yields a noisy measurement recordand whether or not we perform feedback.

Stochastic thermodynamics based on incomplete information: Generalized Jarzynski equality with measurement errors with or without feedback