  We introduce a notion being a $k$-fold Lebesgue function for measure
preserving transformations, where any $2$-fold Lebesgue function is just
ordinary Lebesgue. We discuss how this new metrical isomorphisms invariant of
dynamical systems is related to others classical notions in ergodic theory,
mostly focusing on its spectral aspects. In particular, for transformations
with sufficiently many multiple Lebesgue functions we treat the corresponding
multiple analogs of very well-known problems of Banach and Rokhlin.
