  The detailed fluctuation theorem implies symmetry in the generating function
of entropy production probability. The integral fluctuation theorem directly
follows from this symmetry and the normalization of the probability. In this
paper, we rewrite the generating function by integrating measurements and
evolution into a constructed mapping. This mapping is completely positive, and
the original integral FT is determined by the trace-preserving property of
these constructed maps. We illustrate the convenience of this method by
discussing the eigenstate fluctuation theorem and heat exchange between two
baths. This set of methods is also applicable to the generating functions of
quasi-probability, where we observe the Petz recovery map arising naturally
from this approach.
