  The standard approach to renormalization relies, technically, on the
asymptotic perturbation of Gaussian measures embodied in Feynman diagram
theory. From a mathematical standpoint this is not good enough, because thereby
solving the renormalization problem does not immediately traduce into having a
rigorous construction of the corresponding theory. Here, we start developing an
approach to renormalization based on cylinder measures. After explaining how
renormalization can be mathematically better understood in terms of them, we
argue that a renormalization problem can, under certain hypothesis, be reduced
to that of the corresponding strongly coupled theory, and obtain a family of
solutions for the case of scalar bosons whose interaction Lagrangian does not
contain derivatives. As a further application, we produce an explicit, formal
expression for the cylinder measure of a local field with effectively quartic
interaction at any given, fixed scale, in arbitrary dimension.
