Radon-Nikodym derivatives of finitely additive interval measures taking values in a Banach space with basis

Let X be a Banach space with a Schauder basis {en}, and let \u3a6(I)= 11n en 2bI fn(t)dt be a \ufb01nitely additive interval measure on the unit interval [0, 1], where the integrals are taken in the sense of Henstock\u2013Kurzweil. Necessary and su\ufb03cient conditions are given for \u3a6 to be the inde\ufb01nite integral of a Henstock\u2013Kurzweil\u2013Pettis (or Henstock, or variational Henstock) integrable function f:[0, 1] \u2192 X