We present a weaker version of the Fremlin generalized McShane integral (1995) for functions defined on a $\sigma$-finite outer regular quasi Radon measure space $(S,\Sigma ,𝒯,\mu )$ into a Banach space $X$ and study its relation with the Pettis integral. In accordance with this new method of integration, the resulting integral can be expressed as a limit of McShane sums with respect to the weak topology. It is shown that a function $f$ from $S$ into $X$ is weakly McShane integrable on each measurable subset of $S$ if and only if...