  We introduce a general definition of hybrid transforms for constructible
functions. These are integral transforms combining Lebesgue integration and
Euler calculus. Lebesgue integration gives access to well-studied kernels and
to regularity results, while Euler calculus conveys topological information and
allows for compatibility with operations on constructible functions. We conduct
a systematic study of such transforms and introduce two new ones: the
Euler-Fourier and Euler-Laplace transforms. We show that the first has a left
inverse and that the second provides a satisfactory generalization of Govc and
Hepworth's persistent magnitude to constructible sheaves, in particular to
multi-parameter persistent modules. Finally, we prove index-theoretic formulae
expressing a wide class of hybrid transforms as generalized Euler integral
transforms. This yields expectation formulae for transforms of constructible
functions associated to (sub)level-sets persistence of random Gaussian
filtrations.
