To the Lebesgue theorem on integral differentiation (Q2719446)

The author proposes a generalization of the Lebesgue theorem as follows. Let the function \(\xi(x)\) be such that \(|\xi(x)|(1+\max(0,\ln|\xi(x)|))\) is locally summable on \(R^n\). Then the equality NEWLINE\[NEWLINE \lim\limits_{\Delta x\to 0} \Biggl[\frac{1}{\Delta x} \int_{x_i}^{x_i+\Delta x} \xi(x_1,x_2,\dots,x_{i-1},y,x_{i+1},\dots, x_n) dy\Biggr] = \xi(x) NEWLINE\]NEWLINE holds true for almost all \( x\in R^n\).