Measure on space of maps between circle group and Z2

Question

I am looking for a measure on the set $S$ of all maps $s : \mathbb{T} \to \mathbb{Z}_2$ between the circle group $ \mathbb{T} = \{c \in \mathbb{C} \mid c c^* = 1 \}$ with the ordinary multiplication in $\mathbb{C}$, and the two-element group $\mathbb{Z}_2 = \{1,-1\}$ (represented as sign group with ordinary multiplication). So far, all I got is that $S$ forms itself a (Boolean) group with the pointwise product $(st)(x) = s(x)t(x)$ (but I have no clue how a topology on $S$ could look like).

Is there any canonical way to define a measure on $S$?

Otherwise, if $S$ is not a measurable space, is there a reasonable way to restrict it, say to the set of all measurable functions $\mathbb{T} \to \mathbb{Z}_2$ (with Borel $\sigma$-algebra on $\mathbb{T}$ and power set on $\mathbb{Z}_2$) such that it becomes measurable?

Apologies if the question is trivial. Being a physicist, my lessons on measure theory and topology were long ago and very rudimentary, and Wikipedia and Google haven't brought me much further. Appreciate any helpful keyword.

Answer 1

First of all, a map $s:\mathbb{T}\to\{0,1\}$ does nothing more than to identify a subset of $\mathbb{T}$.

The question would simply be that of asking what measure you want to impose on $\mathbb{T}$:

which subsets are measurable, how to define the measure, etc.

The Lebesgue measure on $\mathbb{R}^2$, restricted to $\mathbb{T}$, would be a standard example.

$\\$

You speak of the group $\mathbb{T}$ and of the two-element group.

Perhaps you want to require $s$ to be a group-homomorphism.

What is $\mathbb{T}$, in terms of group-structure?

If you assume the axiom of choice,

$\mathbb{T}\cong \dfrac{\mathbb{Q}}{\mathbb{Z}}\oplus\langle S\rangle$,

where $S$ is a set having the same cardinality as $\mathbb{R}$ has,

and $\langle S\rangle$ denotes the free abelian group of which the generators are precisely the elements of $S$.

If you want to know why $S$ has the same cardinality as $\mathbb{R}$ does, I refer you to this answer. Take note of the sentence that begins with "Suppose that there was a Hamel basis". If you know enough about cardinalities, you'll be able to work out why $S$ must be precisely as large as such a Hamel basis.

Each group-homomorphism $s:\dfrac{\mathbb{Q}}{\mathbb{Z}}\oplus\langle S\rangle\to\mathbb{Z}_2$ does no more than identify a subset of $S$. Exercise: $s$ has to send $\dfrac{\mathbb{Q}}{\mathbb{Z}}\oplus\{e\}$ to $0$.

The question is, again, what measure you want to impose on $S$.