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Without violating causality, we allow performing measurements in time reverseprocess of a feedback manipulated stochastic system. As a result we come acrossan entropy production due to the measurement process. This entropy production,in addition to the usual system and medium entropy production, constitutes thetotal entropy roduction of the combined system of the reservoir, the system andthe feedback controller. We show that this total entropy production of "full"system satisfies an integrated fluctuation theorem as well as a detailedfluctuation theorem as expected. We illustrate and verify this idea throughexplicit calculation and direct simulation in two examples.

Nonequilibrium fluctuation theorem for systems under discrete and continuous feedback control