  Fluctuation theorems and the second law of thermodynamics are powerful
relations constraining the behavior of out-of-equilibrium systems. While there
exist generalizations of these relations to feedback controlled quantum
systems, their applicability is limited, in particular when considering strong
and continuous measurements. In this letter, we overcome this shortcoming by
deriving a novel fluctuation theorem, and the associated second law of
information thermodynamics, which remain applicable in arbitrary feedback
control scenarios. In our second law, the entropy production is bounded by the
coarse-grained entropy production which is inferrable from the measurement
outcomes, an experimentally accessible quantity that does not diverge even
under strong continuous measurements. We illustrate our results by a qubit
undergoing discrete and continuous measurement, where our approach provides a
useful bound on the entropy production for all measurement strengths.
