  The Minkowski question mark function is a rich object which can be explored
from the perspective of dynamical systems, complex dynamics, metric number
theory, multifractal analysis, transfer operators, integral transforms, and as
a function itself via analysis of continued fractions and convergents. Our
permanent target, however, was to get arithmetic interpretation of the moments
of ?(x) (which are relatives of periods of Maass wave forms) and to relate the
function ?(x) to certain modular objects. In this paper we establish this link,
embedding ?(x) not into the modular-world itself, but into a space of functions
which are generalizations and which we call mean-modular forms. For this
purpose we construct a wide class of measures, and also investigate modular
forms for congruence subgroups which additionally satisfy the three term
functional equation. From this perspective, the modular forms for the whole
modular group as well as the Stieltjes transform of ?(x) (the dyadic period
function) minus the Eisenstein series of weight 2 fall under the same uniform
definition. The main result is the construction of the canonical isomorphism
between the spaces of quasi-modular forms and mean-modular forms. This gives
unexpected Minkowski question mark function-related interpretation of
quasi-modular forms.
