I am looking for good references (survey, monograph, or paper with a solid background section) on the Banach space / functional analytic structure of spaces of finite signed measures $\mathcal{M}(X)$, primarily for $X = \mathbb{R}$, $\mathbb{R}^2$, or more generally $\mathbb{R}^d$ (and possibly Polish spaces).
I am interested in understanding these spaces as Banach spaces in their own right—particularly the geometry and topology relevant to studying operators acting on them. The following topics are of particular interest:
- The tensor product structure, e.g. the canonical identification $M(\mathbb{R}) \widehat{\otimes}_\pi M(\mathbb{R}) \cong M(\mathbb{R}^2)$, and its generalizations to higher dimensions.
- Weak and weak* topologies on $\mathcal{M}(X)$, including compactness and tightness criteria.
- Properties of linear operators on these spaces and their continuity under the relevant topologies.
- Structural questions such as whether the subset of measures of the form $\sum_{i=1}^k \prod_{j=1}^d \mu_{i,j}$ (finite “rank” measures) is closed in suitable norms or topologies.
I am also interested, more secondarily, in the behavior of probability measures on $\mathcal{M}(X)$, particularly differences of probability measures and how these behave under linear transformations, as well as transform methods (Fourier, Laplace) on these spaces.
For background, I am currently using:
- Folland, Real Analysis
- Kallenberg, Foundations of Modern Probability
- Fabian et al., Functional Analysis and Infinite-Dimensional Geometry
I have also looked at Ryan's Introduction to Tensor Products of Banach Spaces, though it seems to address a somewhat different setting.
I understand this is a bit vague, but I am mainly trying to get a sense of what functional-analytic or Banach space tools are available for studying $\mathcal{M}(X)$ in this way.
Edit: I am also interested in settings where $X$ is a compact subset of $\mathbb{R}$ or $\mathbb{R}^d$, we could just assume that set is the unit interval or cube.
