  The purpose of this paper is to construct a new class of separable Banach
spaces $\K^p[\mathbb{B}], \; 1\leq p \leq \infty$. Each of these spaces contain
the $ \mcL^p[\mathbb{B}] $ spaces, as well as the space $\mfM[\R^\iy]$, 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 $\K^p[\mathbb{B}].$
