The integration with respect to a vector measure may be applied in order to approximate a function in a Hilbert space by means of a finite orthogonal sequence attending to two different error criterions. In particular, if is a Lebesgue measurable set, , and is a finite family of disjoint subsets of , we can obtain a measure and an approximation satisfying the following conditions: (1) is the projection of the function in the subspace generated by in the Hilbert space . (2) The integral distance between and on the sets is small.
L. M. García-Raffi. D. Ginestar. E. A. Sánchez-Pérez. "Integration with respect to a vector measure and function approximation." Abstr. Appl. Anal. 5 (4) 207 - 226, 2000. https://doi.org/10.1155/S1085337500000221