Kuelbs-Steadman spaces on Separable Banach spaces

The purpose of this paper is to construct a new class of separable Banach spaces ${\text{K}}^p[\mathbb{B}],1\le p\le \mathrm{\infty }$. Each of these spaces contain the ${\text{mcL}}^p[\mathbb{B}]$ spaces, as well as the space $\text{mfM}[{\mathbb{R}}^{\text{iy}}]$, of finitely additive measures as dense continuous compact embeddings. These spaces are of interest because they also contain the Henstock-Kurzweil integrable functions on $\mathbb{B}$. Finally, we offer a interesting approach to the Fourier transform on ${\text{K}}^p[\mathbb{B}]\mathrm{.}$