$a$

Our task in this post will be to develop the basic theory and notation for inframeasures and sa-measures. The proofs and concepts require some topology and functional analysis. We assume the reader is familiar with topology and linear algebra but not functional analysis, and will explain the functional analysis concepts more. If you wish to read through these posts, PM me to get a link to MIRIxDiscord, we'll be doing a group readthrough where I or Vanessa can answer questions. Here's the previous post and here are the proof sections. Beware, the proof sections are hard.

Notation Reference

Feel free to skip this segment and refer back to it when needed. Duplicate the tab, keep one of them on this section, and you can look up notation here.

$X, Y$ : some compact metric space. Points in this are denoted by $x$ or $y$ .

$d$ : Some distance metric.

$M^{\pm} (X)$ : The topological vector space of finite signed measures over $X$ equipped with the weak topology. A sequence of measures $m_{n}$ converges to m in the weak topology iff, for all bounded continuous functions f of type $X \to R$ , $\int_{X} f (x) d m_{n}$ limits to $\int_{X} f (x) d m$ . Elements are denoted by $m$ . By Jordan decomposition, we can uniquely split it into $m^{+} + m^{-}$ where the former is all-positive and the latter is all-negative.

$C (X), C (X, [0, 1])$ : The Banach space of continuous functions $X \to R$ . The latter one is the space of continuous functions bounded in $[0, 1]$ . Elements of $C (X, [0, 1])$ are typically denoted by $f$ .

$m (f)$ : We can interpret signed measures as continuous linear functionals on $C (X)$ . This is given by $\int_{X} f (x) d m$ . If $m$ was actually a probability distribution, this would just be $E_{μ} (f)$ . They're generalized expectations.

$b$ : used to refer to the number component of an a-measure or sa-measure.

$M^{s a} (X)$ : The closed convex cone of sa-measures. An sa-meaure is a pair $(m, b)$ where $b + m^{-} (1) \geq 0$ . Elements of this (sa-measures) are denoted by $M$ .

$f^{+}$ : A positive functional. A continuous linear functional that's nonnegative for all sa-measures.

$B$ : A set of sa-measures.

$E_{B} (f)$ : The expectation of a function $f$ relative to a set of sa-measures. Defined as ${inf}_{(m, b) \in B} (m (f) + b)$ .

$B^{min}, B^{u c}$ : The set of minimal points of $B$ (points that can't be written as a different point in $B$ plus a nonzero sa-measure), and the upper...