Let {X ni , u n ≤ i ≤ ν n , n ≥ 1} and {A ni , u n ≤ i ≤ ν n , n ≥ 1} be two arrays of random variables. A new notion of integrability for an array of random variables {X ni } with respect to an array of random variables {A ni } is introduced, weak laws of large numbers for the randomly weighted sums equation are obtained. Our results extend and sharpen the known results in the literature.