To generalize the concept of classic limit of sequence, this paper introduces the definition of almost convergent sequence and proves several important properties together with a necessary and sufficient condition of almost convergence, hence building an equivalent relation between almost and strictly convergence. Moreover, based on the Lebesgue measure and by introducing the conception of density of subsets on {\mathbf{R}}^n,we also provide the definition of almost convergence of measurable functions, including some properties and a basic theorem of almost convergence similar to sequence. Then we introduce the definition of almost continuous function. At last, based on the Lebesgue differential theorem, it is proved that any measurable function is almost continuous, almost everywhere on {\mathbf{R}}^n.