On a criterion for the continuity of the Rayleigh-Ritz operator

The work considers the Rayleigh-Ritz operator identified on the set of pairs of measurable functions that equals to the ratio of their modules if a denominator is different from zero, and zero otherwise. The issue of the continuity of this operator regarding the convergence in measure is studied. It is shown that for the convergence of the value of operator on a sequence of pairs to the value on the limit pair of fonctions, it is necessary not only the convergence in measure of its arguments, but also the convergence in measure of the second argument to the carrier of its limit.