  This paper provides a compact method to lift the free exponential
construction of Mellies-Tabareau-Tasson over the Hyland-Schalk double glueing
for orthogonality categories. A condition ``reciprocity of orthogonality'' is
presented simply enough to lift the free exponential over the double glueing in
terms of the orthogonality. Our general method applies to the monoidal category
TsK of the s-finite transition kernels with countable biproducts. We show (i)
TsK^op has the free exponential, which is shown to be describable in terms of
measure theory. (ii) The s-finite transition kernels have an orthogonality
between measures and measurable functions in terms of Lebesgue integrals. The
orthogonality is reciprocal, hence the free exponential of (i) lifts to the
orthogonality category O_I(TsK^op), which subsumes Ehrhard et al's
probabilistic coherent spaces as the full subcategory of countable measurable
spaces. To lift the free exponential, the measure-theoretic uniform convergence
theorem commuting Lebesgue integral and limit plays a crucial role as well as
Fubini-Tonelli theorem for double integral in s-finiteness. Our
measure-theoretic orthogonality is considered as a continuous version of the
orthogonality of the probabilistic coherent spaces for linear logic, and in
particular provides a two layered decomposition of Crubille et al's direct free
exponential for these spaces.
