The main purpose of this paper is to introduce and investigate the concepts of lacunary strong summability of order $\alpha$ and lacunary statistical convergence of order $\alpha$ of real-valued functions which are measurable (in the Lebesgue sense) in the interval $(1,\infty)$. Some relations between lacunary statistical convergence of order $\alpha$ and lacunary strong summability of order $\beta$ are also given.