Extrapolation in Grand Lebesgue Spaces with $A_{\infty}$ Weights

Results on extrapolation with $A_{\infty}$ weights in grand Lebesgue spaces are obtained. Generally, these spaces are defined with respect to the product measure $\mu_1\times \dotsb\times \mu_n$ on $X_1\times \dotsb\times X_n$, where $(X_i,d_i,\mu_i)$, $i=1,\dots,n$, are spaces of homogeneous type. As applications of the obtained results, new one-weight estimates with $A_{\infty}$ weights for operators of harmonic analysis are derived.