There are two candidates I know of for uniform upper probabilities on $\wp\omega$. The first is the usual relative frequency and the second is a function which I do not know the name of but I read somewhere years ago (Peter Walley 1991)? it is somehow better as a uniform measure on $\wp\omega$, presumably meaning it has better properties is some sense. Letting $I\subseteq\omega$ denote its own indicator function
Frequency -- $\overline{Q}:\wp\omega\rightarrow\left[0,1\right]:I\mapsto\overline{Q}[I]=\lim_{n\to\infty}\sup_{k\in\omega}\frac{1}{n+k+1}\sum_{i=0}^{n+k}I(i)$
Uniform -- $\overline{U}:\wp\omega\rightarrow\left[0,1\right]:I\mapsto\overline{U}[I]=\lim_{n\to\infty}\sup_{k\in\omega}\frac{1}{n+1}\sum_{i=k}^{n+k}I\left(i\right)$
$\overline{Q}$ is well-defined because it's a $\limsup$ and $\overline{U}$ is well defined by Fekete's Subadditive Lemma. There are many definitions of upper probability, and/or the dual lower probabilities $\underline{m}\left[I\right]=1-\overline{m}\left[I^{\complement}\right]$. Some of the typical properties they may have are $0\le\overline{m}\left[I\right]\le1$, $\overline{m}\left[\varnothing\right]=0$, $\overline{m}\left[\omega\right]=1$, $\overline{m}\left[I\right]\le\overline{m}\left[J\right]$ when $I\subseteq J$ (monotone) and $\overline{m}\left[I\cup J\right]\le\overline{m}\left[I\right]+\overline{m}\left[J\right]$ (sub-additive). The above functions have these properties and more.
Can anyone shed any light on the uniform distribution $\overline{U}$ above in terms of it's name, properties, relationship to $\overline{Q}$, and where it crops up and in what role in mathematics. I know this is a very broad question but I have always wondered about it. In particular does $\overline{Q}=\overline{U}$, and if so how do you prove it, and if not when not?
SOME BACKGROUND
While I haven't found a direct answer to my question on the relationship between $\overline{Q}$ and $\overline{U}$ I have done some research. Banach Limits are a topic in Functional Analysis that covers functionals like these. Meyer Jerison wrote several papers 2009 and 1957 on this. There is another topic called statistical limits developed in apparent isolation by J.A. Fridy.
Using notation from Functional analysis $l_{\infty}$ is the set of bounded real sequences and $c\subseteq l_{\infty}$ is the subset that converge. Note $l_{\infty}$ therefore includes the characteristic function of every $I\subseteq\omega$. The limit functional $\lim:c\to\mathbb{R}:x\mapsto\lim\left(x\right)$ is a linear functional with well known properties. A Banach limit is an extension of $\lim$ to $l_{\infty}$ that preserves these properties. Let $\mathfrak{B}$ be the set of Banach limits then $L\in\mathfrak{B}$ means $L:l_{\infty}\to\mathbb{R}$ and
- Extension: If $x\in c$ then $L\left(x\right)=\lim\left(x\right)$.
- Linear: $L\left(ax+by\right)=aL\left(x\right)+bL\left(y\right)$.
- Positive: If $x\ge0$ then $L\left(x\right)\ge0$.
- Norm: $L\left(\underline{1}\right)=1$ where $\underline{1}:\omega\to\mathbb{R}:i\mapsto1$ (constant).
- Shift Invariant: $L\left(x\circ s\right)=L\left(x\right)$ where $s:\omega\to\omega:i\mapsto i+1$ (successor).
The theory is quite extensive and deep, and there are several ways to prove Banach Limits exist including direct examples, in fact Banach Limits have been characterised in several ways.
Recall that given any filter $\mathcal{F}$ on $\omega$ then $a\in\mathbb{R}$ is an $\mathcal{F}\text{-}\lim$ of the sequence $x\in\mathbb{R}^{\omega}$ means $x^{-1}U\in\mathcal{F}$ for every neighborhood $U$ of $a$. When the range of $x$ is confined to a compact subset $B\subseteq\mathbb{R}$ and $\mathcal{U}$ is an ultrafilter the $\mathcal{U}\text{-}\lim$ always exists. Clearly if $\mathcal{F}\subseteq\mathcal{G}$ and $\mathcal{F}\text{-}\lim x$ exists then $\mathcal{G}\text{-}\lim x=\mathcal{F}\text{-}\lim x$.
Ordinary convergence is based on the Frechet filter $\mathcal{N}:=\left\{ I\subseteq\omega\mid I^{\complement}\text{ finite}\right\}$ and Fridy's statistical limit is based on the filter $\mathcal{Q}:=\left\{ I\subseteq\omega\mid\frac{1}{n+1}\sum_{i=0}^{n}I(i)\longrightarrow1\right\}$ .
An example of a Banach limit is that if $\mathcal{U}$ is an ultrafilter on $\omega$ then $J\left(x\right)=\mathcal{U}\text{-}\lim\frac{1}{n+1}\sum_{i=0}^{n}x_{i}$ is a Banach limit, and even more generally if $p\in\omega^{\omega}$ is any sequence in $\omega$ then $J\left(x\right)=\mathcal{U}\text{-}\lim\frac{1}{n+1}\sum_{i=k}^{n+k}x_{p_{i}}$ is a Banach limit.
Jerison shows that the set of Banach limits is a closed, compact, convex subset of the unit ball in the dual space $l_{\infty}^{*}$ in the weak* topology. If we say $a\in\mathbb{R}$ is some kind of limit of $x$ when $L\left(x\right)=a$ for some $L\in\mathfrak{B}$ then $\mathfrak{L}\left(x\right):=\left\{ L\left(x\right)\mid L\in\mathfrak{B}\right\}$ contains every real that is some kind of limit of $x$. Jerison shows that $\mathfrak{L}\left(x\right)=\left[\underline{M}\left(x\right),\overline{M}\left(x\right)\right]$ where $$\underline{M}\left(x\right) := \lim_{n\rightarrow\infty}\left(\liminf T_{n}\left(x\right)\right)$$
$$\overline{M}\left(x\right) := \lim_{n\rightarrow\infty}\left(\limsup T_{n}\left(x\right)\right)$$ and $T_{n}\left(x\right)$ is the sequence defined by $$T_{n}\left(x\right):\omega\to\mathbb{R}:k\mapsto\frac{1}{n+1}\sum_{i=k}^{k+n}x_{i}$$ There is a linkage here to the $\overline{Q}$ and $\overline{U}$ in my question because $$\overline{Q}\left[I\right] = \lim_{n\to\infty}\sup_{k\in\omega}T_{n+k}\left(I\right)_{0}$$ $$\overline{U}[I] = \lim_{n\to\infty}\sup_{k\in\omega}T_{n}\left(I\right)_{k}$$
Unfortunately I can't see how this throws much light on my original question which is more along the lines of when does $\overline{Q}[I]\neq\overline{U}[I]$.