I2-lacunary strongly summability for multidimensional measurable functions

The concept of a statistical convergence was introduced by Fast [9], and Steinhaus [30] independently in the same year 1951. Actually, the idea of statistical convergence was used to proved theorems on the statistical convergence of Fourier series by Zygmund [31] in the first edition of his celebrated monograph published in Warsaw. He used the term “almost convergence” place of statistical convergence and at that time this idea was not recognized much. Since the term “almost convergence” was already in use Lorentz [18], Fast [9] had to choose a different name for his concept and “statistical convergence” was mostly the suitable one. Active research on this topic started after the paper of Fridy [10] and since then a large collection of literature has appeared. At the last quarter of the 20th century, statistical convergence has been discussed and captured important aspect in creating the basis of several investigations conducted in main branches of mathematics such as the theory of number [7], measure theory [19], trigonometric series [31], probability theory [6], and approximation theory [12]. In addition, it was further investigated from the sequence space point of view and linked with summability theory by Connor [4], Et at. al. [8], Kolk [13], Orhan et al. [11], Kumar and Mursaleen [15], Rath and Tripathy [24], Šalát [25], and many others made substantial contributions to the theory. Definition 1.1. Let R be a subset of N and Rm = {i 6 m : i ∈ R}. The natural density of R is defined δ(R) = limm 1 m |Rm| provided it exists. Here, and in