Approach to the construction of the spaces $ S{D^p}[\mathbb{R}^\infty]$ for $1 \leq p \leq \infty$

TL;DR

This paper develops a method to construct separable Banach spaces that include various function spaces and measures, enhancing the mathematical framework for analysis on infinite-dimensional real spaces.

Contribution

It introduces a novel construction of Banach spaces $SD^p[ e^ infty]$ that embed $L^p$ spaces, measures, and integrable functions in a unified framework.

Findings

Constructed separable Banach spaces containing $L^p$ spaces.

Embedded finitely additive measures as dense subsets.

Included Henstock-Kurzweil integrable functions within the spaces.

Abstract

The objective of this paper is to construct separable Banach spaces $S{D^p}[\mathbb{R}^\infty]$ for $1\leq p \leq \infty$, each of which contains the $L^p[\mathbb{R}^\infty] $ spaces, as well as finitely additive measures, as compact dense embedding. Also these spaces contains Henstock-Kurzweil integrable functions.