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We analyze the complex phenomenology of the Non-Classical Rotational Inertia(NCRI) observed at low temperature in solid $^4$He within the context of a twodimensional Berezinski-Kosterlitz-Thouless transition in a premelted $^4$Hefilm at the grain boundaries. We show that both the temperature and $^3$Hedoping dependence of the NCRI fraction (NCRIF) can be ascribed to finite sizeeffects induced by the finite grain size. We give an estimate of the averagesize of the grains which we argue to be limited by the isotopic $^3$Heimpurities and we provide a simple power-law relation between the NCRIF and the$^3$He concentration.

Finite size Berezinski-Kosterlitz-Thouless transition at grain boundaries in solid $^4$He and role of $^3$He impurities