On the behavior of integrable functions at infinity

We investigate the behavior of sequences $(f(c_{n}x))$ for Lebesgue integrable functions $f:{\mathbb{R}}^d\to \mathbb{R}$. In particular, we give a~description of classes of multipliers $(c_n)$ and $(d_n)$ such that $f(c_{n}x)\to 0$ or ${\sum }_{n=1}^{\mathrm{\infty }}\mathrm{\mid }f(d_{n}x)\mathrm{\mid }<\mathrm{\infty }$ for $\lambda$ almost every $x\in\mathbb R^d$.