We introduce a natural and more flexible approach to the definition of vector valued integral that will completely forgo any measurability assumption, strengthen the existing various classical concepts of integral, and provide a continuous thread tying the subject matter together. As applications, we obtain extensions of the Lebesgue convergence theorems, the Dvoretsky-Rogers theorem, and the Orlicz-Pettis theorem.

vector valued Henstock-Kurzweil integral, Lebesgue-Bochner integral.