I have heard that nonrenormalizable operators (i.e., mass dimension greater than 4) can be "induced" in the Lagrangian (that we started with) via loop effects. However, I do not understand what does it mean by a new operator or term in the Lagrangian being induced.
In QED, loops generally correspond to self-energy diagrams, vacuum polarization diagrams and vertex correction diagrams which modify the bare mass, bare charge, and bare fields to the corresponding renormalized quantities by adding counterterms (or by splitting the original Lagrangian into a renormalized part and a counterterm part). Counterterms must be included, as I understand, to get rid of various divergences (or cut-off dependence).
$\bullet$ But why should one include nonrenormalizable terms in the original Lagrangian (in which none of the terms did resemble the induced operator)?
$\bullet$ Can they be thought of as counnterterms?
$\bullet$ How many of them can/should be included?
$\bullet$ Can one clarify the concept inducing non-renormalizable operators in the context of a simple field theory?