I am reading the following paper
Beloso-Herves, C. and Monteiro, P.K. Information and s-algebras. Economic Theory, 54(2): 405-418, 2013.
and have a question of a technical nature which the authors do not define/explain. The paper uses equivalence classes, partitions of sets, and $\sigma$-fields as the foundation stone of their research. They utilize a number of definitions that might be needed, or perhaps my questions in bold can be answered in isolation without any of this context.
Some definitions
Partitions of $S$: $S_{\tau}(S)$ is a partition of $S$ if
- $S:=\cup\left\{A\subseteq S:A\in S_{\tau}(S)\right\}$
- $A_{1},A_{2}\in S_{\tau}(S)$ and $A_{1}\not =A_{2}$ imply $A_{1}\cap A_{2}=\emptyset$.
I have changed their notation to use $S_{\tau}$ (the authors just use $\tau$).
Atoms of a class: If $\mathcal{A}$ is a class of subsets of $S$, a set $\text{atom}(\mathcal{A})\in\mathcal{A}$ is an atom of $\mathcal{A}$, if for all $A\in\mathcal{A}$ satisfying $A\subseteq \text{atom}(\mathcal{A})$, $A=\emptyset$ or $A=\text{atom}(\mathcal{A})$. For any $s\in S$, $\text{atom}(s,\mathcal{A})$ will be used to denote the atom of $\mathcal{A}$ to which $s$ belongs.
A signal on $S$: A mapping $f:S\longrightarrow T$ is a signal on $S$ with values on $T$
Partition induced by a signal: The partition $\tau_{f}$ of $S$ induced by a signal $f:S\longrightarrow T$ is
$$\tau_{f}=\left\{f^{-1}(t):t\in T\right\}$$
Suppose we have an equivalence relation $\sim$ on $S$ generating the quotient set $S_{\sim}:=S\backslash\sim:=\{[s]_{\sim}:s\in S\}$. We need the following definition of the natural projection (so called by the authors);
The natural projection:
\begin{align} \phi_{\sim}:\hspace{5pt}S&\longrightarrow S_{\sim}\\ \phi_{\sim}(s)&\mapsto [s]_{\sim} \end{align}
Finally we need definitions of functions;
Function definition
A function $f:S\longrightarrow T$ requires $f(s)=f(s')$ to hold whenever $s=s'$, or equivalently as per Jech p11 (Set Theory, 2003, 3rd edition) if $R$ is a binary relation on $S\times T$ then $R$ is a function if $(s,t),(s,u)\in R$ imply $t=u$.
Example function for $\phi_{\sim}$
$R_{\phi_{\sim}}$ is the set-theoretic version of $\phi_{\sim}$;
\begin{align*} R_{\phi_{\sim}}=\left\{(s,B)\in S\times S_{\sim}:\phi_{\sim}(s)=B\right\} \end{align*}
which does indeed satisfy $(s,B),(s,C)\in R_{\phi_{\sim}}$ implies $B=C$: since $B,C\in S_{\sim}$ then $B=[b]_{\sim}$ and $C=[c]_{\sim}$ for some $b,c\in S$ and hence $(s,B),(s,C)\in R_{\phi_{\sim}}$ implies $\phi_{\sim}(s)=[b]_{\sim}$ and $\phi_{\sim}(s)=[c]_{\sim}$. But $s\in [s]_{\sim}$ and so $\phi_{\sim}(s)=[s]_{\sim}$. Thus $[s]_{\sim}=[b]_{\sim}=[c]_{\sim}$ holds, or equivalently $[s]_{\sim}=B=C$.
Jech (2003) calls the more familiar (to me anyway) mapping notation $\phi$ the "standard" notation which are obviously linked in the following way;
\begin{align*} \forall (s,B)\in S\times S_{\sim}:\hspace{10pt}(s,B)\in R_{\phi_{\sim}}\Longleftrightarrow \phi_{\sim}(s)=B \end{align*}
An equivalence relation on an arbitrary partition $S_{\tau}$
Suppose $S_{\tau}$ is an arbitrary partition of $S$. Using similar notation as for members of quotient sets, for any $s\in S$ let $[s]_{\tau}\in S_{\tau}$ be the set of the partition that $s$ lies in. Define the following equivalence relation on $S_{\tau}$;
\begin{align*} s\sim_{\tau} t\Longleftrightarrow \text{atom}(s,S_{\tau})=\text{atom}(t,S_{\tau}) \end{align*}
Choose $[s]_{\sim_{\tau}}\in S_{\sim_{\tau}}$ and any $t\in [s]_{\sim_{\tau}}$ so that from above $s\in\text{atom}(s,S_{\tau})=\text{atom}(t,S_{\tau})\ni t$ is true. Since $t$ was arbitrary then $[s]_{\sim_{\tau}}\subseteq\text{atom}(s,S_{\tau})$ must be true, so that $[s]_{\sim_{\tau}}\not=\emptyset$ and the definition of atoms imply
$$[s]_{\sim_{\tau}}=\text{atom}(s,S_{\tau})$$
Now choose $[s]_{\tau}\in S_{\tau}$ and suppose $[s]_{\tau}\not=\text{atom}(s,S_{\tau})$. Then the properties of partitions implies $[s]_{\tau}\cap\text{atom}(s,S_{\tau})=\emptyset$, which contradicts $s\in[s]_{\tau}$ and $s\in\text{atom}(s,S_{\tau})$, which implies
$$[s]_{\tau}=\text{atom}(s,S_{\tau})$$
Accordingly we have
\begin{align} S_{\sim_{\tau}}&=\{[s]_{\sim_{\tau}}:s\in S\}\hspace{100pt}(1)\\ &=\{\text{atom}(s,S_{\tau}):s\in S\}\\ &=\{[s]_{\tau}:s\in S\}\\ &=S_{\tau} \end{align}
My question
The author's claim the following:
- For any signal $f:S\longrightarrow T$ then $\tau_{f}$ is a partition of $S$
- A general partition $S_{\tau}$ of $S$ is induced by some signal $f:S\longrightarrow T$ (namely $f=\phi_{\sim_{\tau}}$)
Whilst I am happy $\tau_{f}$ does indeed satisfy the above definition of a partition of $S$, I am less sure of the second claim for the simple reason the definition of $\tau_{f}$ depends on $\phi_{\tau}^{-1}$ which does not exist given $\phi_{\tau}$ is not 1-1: for all $[s]_{\sim_{\tau}}\in S_{\sim_{\tau}}$ if there exists a $t\in [s]_{\sim_{\tau}}$ such that $s\not=t$ then $\phi_{\tau}(s)\mapsto [s]_{\sim_{\tau}}=\phi_{\tau}(t)$. However if $\phi_{\tau}^{-1}$ is meant to mean inverse images of sets in $S_{\sim_{\tau}}$ then the claim is easy to prove. Firstly by inverse images I mean
\begin{align*} \phi_{\tau}^{-1}([s]_{\sim})&=\left\{t\in S:\phi_{\tau}(t)\in [s]_{\sim_{\tau}}\right\}\hspace{100pt}(2)\\ &=\left\{t\in S:[t]_{\sim_{\tau}}\in [s]_{\sim_{\tau}}\right\}\\ &=\left\{t\in S:[t]_{\sim_{\tau}}= [s]_{\sim_{\tau}}\right\}\\ &=\left\{t\in S:t\sim_{\tau} s\right\}\\ &=[s]_{\sim_{\tau}}\hspace{50pt}\text{ for all } [s]_{\sim_{\tau}}\in S_{\sim_{\tau}} \end{align*}
So given $\phi_{\tau}:S\longrightarrow S_{\sim_{\tau}}$, applying the definition of $\tau_{\phi}$ we have
\begin{align*} \tau_{\phi_{\tau}}&=\{\phi_{\tau}^{-1}([s]_{\sim_{\tau}}):[s]_{\sim_{\tau}}\in S_{\sim_{\tau}}\}\\ &=\{[s]_{\sim_{\tau}}:[s]_{\sim_{\tau}}\in S_{\sim_{\tau}}\}\\ &=\{[s]_{\sim_{\tau}}:s\in S\}\\ &=S_{\sim_{\tau}}\\ &=S_{\tau} \end{align*}
where the last line follows from (1)
Whilst the above proof works (for me at least), I am uncertain about switching between $\phi_{\tau}:S\longrightarrow S_{\sim_{\tau}}$ which is a surjective mapping (a function) and $\phi_{\tau}^{-1}$ which is not a function but rather a set function since it maps sets in it's domain $S_{\sim_{\tau}}$ to the same sets, i.e. $\phi_{\tau}^{-1}:S_{\sim_{\tau}}\longrightarrow S_{\sim_{\tau}}$ (however if we "view" the equivalence classes as points and not sets, then perhaps we can claim $\phi_{\tau}^{-1}$ is in fact a 1-1 function)? My concern is not alleviated when for a general mapping $f:S\longrightarrow T$ I see on p11 Jech (2003) they use $f_{-1}$ to denote and inverse image and reserve $f^{-1}$ for a 1-1 function where $f^{-1}(s)=t$ if and only if $s=f(t)$.
So I suppose after all this my question boils down simply to this: Is it OK to use a function $\phi_{\tau}:S\longrightarrow S_{\sim_{\tau}}$ to do one thing (in this case be able to map points of $S$ to their equivalence classes in $S_{\sim_{\tau}}$), but to then ignore the fact that $\phi_{\tau}^{-1}:S_{\sim_{\tau}}\longrightarrow S_{\sim_{\tau}}$ is a set function and to use it as an inverse image to do something else that depends on the first construct that $\phi_{\tau}$ is a (standard) function (in this case to construct $\tau_{\phi_{\tau}}$)? Maybe the answer is "it does not matter" as long as the reader recognises $\phi_{\tau}^{-1}$ is not a function? As ever in mathematics we could strive for 100% clarity and write $\phi_{\tau,-1}$ in (2) instead of $\phi_{\tau}^{-1}$ but "no harm is done" if not?
In contrast I have had no problem with the following set-up in measure theory where $\mu:\mathscr{S}\longrightarrow [0,1]$ is a probability measure on a $\sigma$-field of subsets of $S$, and $X:\Omega\longrightarrow S$ is a random variable with law $\mu$ where $Pr\{X\in B\}=X^{-1}(B)=\mu(B)$ for all $B\in\mathscr{S}$ where $X^{-1}(B)$ is the inverse image;
\begin{align*} X^{-1}(B)=\left\{\omega\in\Omega:X(\omega)\in B\right\} \end{align*}
Here $X$ is a function from $\Omega$ to $S$ whereas $X^{-1}:\mathscr{S}\longrightarrow \Omega$ is a set function. In Billingsley (Probability and Measure, 1995, 3rd Edition) which I use for measure theory I see no mention of $X^{-1}$ being written $X_{-1}$ to make clear this distinction. In fact I never consciously noticed this "mixing" of functions and inverse images was happening, since I never made the mistake of assuming $X^{-1}$ was a function.