Abstract: For a physics systerm which exhibits memory,if memory is preserved only at points of random self-similar fractals,we define random memory functions and give the connection between the expecta- tion of flux and the fractional integral.In particular,when memory sets degenerate to Cantor type frac- tals or non-random self-similar fractals our results coincide with that of Nigmatullin and Ren et al..

Key words: random self-similar fractals, memory functions, memory measures, Laplace transform ,