This question is followup to the [previous similar question](https://mathoverflow.net/questions/330088/proving-that-preorder-on-the-set-of-measurable-functions-is-symmetric/). There I was trying to find good sufficient condition for abstract preorder to be symmetric, but now, as I have found good formalization of my specific preorder, I'll better ask about it directly.


Let's say we have arbitrary finite measure space $\langle \Omega, \mathfrak{I}, \mathbb{P} \rangle$. For any functions $\rho : Y \rightarrow X$ and measurable $f : \Omega \rightarrow X$ with any finite $X$ and $Y$ we define $H\_{\rho}(f) = \{\mathbb{P} \circ g^{-1} \mid f = \rho \circ g, g \text{ is measurable}\}$ – set of functions $Y \rightarrow \mathbb{R}\_{\ge 0}$ characterizing internal structure of $f$ induced by $\rho$. Preorder $\precsim$ is defined on set of measurable functions $\{ f : \Omega \rightarrow X \}$ as follows – $f\_1 \precsim f\_2 \iff H\_{\rho}(f\_1) \subseteq H\_{\rho}(f\_2), \forall \rho$.


I believe that such preorder is symmetric ($f\_1 \precsim f\_2 \Leftrightarrow f\_2 \precsim f\_1, \forall f\_1, f\_2$) due to external reasons (it has respective game-theoretical interpretation), but I'm unable to prove it as defined. Can anyone help me, please?


UPDATE: Please, take note, $\forall \rho$ here means for all $\rho$ across every possible finite $Y$, not just one externally fixed.


UPDATE: To keep you guys interested and maybe bootstrap thoughts, I'll add some facts.


1. $f\_1 \precsim f\_2 \Rightarrow \xi \circ f\_1 \precsim \xi \circ f\_2, \forall f\_1, f\_2, \xi$ – this follows trivially from definition of $H\_{\rho}$.
2. Consequentially, symmetry of $\precsim$ is easily provable for $f\_2 = \pi \circ f\_1, \pi : X \leftrightarrow X$. Because of $f\_1 \precsim f\_2 \Rightarrow \pi \circ f\_1 \precsim \pi \circ f\_2$ we can build infinite chain of $f\_1 \precsim \pi \circ f\_1 \precsim \pi \circ \pi \circ f\_1 \precsim \ldots$. Combinatorics says that there must be finite cycle $\pi \circ ... \circ \pi$ synonymous to identity permutation, which leads to $f\_2 = \pi \circ f\_1 \precsim \pi \circ ... \circ \pi \circ f\_1 = f\_1$.
3. Consequentially, symmetry of $\precsim$ is easily provable for measure spaces with finite algebra. In such case any measurable function $f\_i$ can be derived from singular function $g$, that transforms each atom to its own element of codomain, as $f\_i = \xi\_i \circ g$. For such $g$ and its permutations $g \precsim \pi \circ g \Leftrightarrow \pi \circ g \precsim g \Leftrightarrow \mathbb{P} \circ g^{-1} = \mathbb{P} \circ (\pi \circ g)^{-1}$ which leads to $f\_1 \precsim f\_2 \Leftrightarrow f\_2 \precsim f\_1$ by definition of $H\_{\rho}$.


Unfortunately, here my progress bumps into wall – going from this subcases to generic case seems to be beyond my knowlegde. And this is sad, because symmetry of preorder in question is a cornerstone of formal argumentation for very interesting new result in game theory. I would be very grateful even for slightest hints.



Again, proving that specific preorder on the set of measurable functions is symmetric
I did it, finally. Without going into much detail, it is possible to prove following fact - if there exists at least one pair of measurable $f\_1, f\_2 : \Omega \rightarrow X, f\_1 \prec f\_2 \equiv f\_1 \precsim f\_2 \cap \neg (f\_1 \succsim f\_2)$, then for any measurable $g\_1 : \Omega \rightarrow Y$ it is possible to construct measurable $g\_2 : \Omega \rightarrow Y$ for which $g\_1 \prec g\_2$. This is not possible though, because space of measurable functions with finite codomain is compact in Tychonoff topology, so for any partial order there must be at least one maximal element.


UPDATE: For people interested here is line of reasoning:


1. I introduce pseudometric $d(f\_1, f\_2) = \mathbb{P}(f\_1 \ne f\_2)$ for space of $\Omega \rightarrow X$ and metric $d(\mu\_1, \mu\_2) = \max \lvert \mu\_1 - \mu\_2 \rvert$ for space of $Y \rightarrow \mathbb{R}\_{\ge 0}$. Under this $H\_\rho(f)$ is continuous in Hausdorff sense, so $H\_\rho^{-1}(\mu) = \{\text{measurable } f \mid \mu \in H\_\rho(f)\}$ is continuous too.
2. Consequentially, if $f\_1 \precsim f\_2$, then for any measurable $g\_1$ such that $f\_1 = \rho \circ g\_1$ it is possible to construct measurable $g\_2$ such that $f\_2 = \rho \circ g\_2$ and $g\_1 \precsim g\_2$. Proof here uses triviality of building $g\_2$ which $H\_\rho(g\_2)$ contains any finite subset of $H\_\rho(g\_1)$.
3. Because of $f\_1 \precsim f\_2 \Rightarrow \rho \circ f\_1 \precsim \rho \circ f\_2$ previous statement holds for strict relation $\prec$ too.
4. Supposing $f\_1 \prec f\_2 : \Omega \rightarrow X$, for any $g\_1 : \Omega \rightarrow Y$ we build $h\_1 = (f\_1, g\_1) : \Omega \rightarrow X \times Y$ and $\rho(x, y) = x, \xi(x, y) = y$. As $f\_1 = \rho \circ h\_1$ there exists measurable $h\_2$ such that $f\_2 = \rho \circ h\_2$ and $h\_1 \prec h\_2$. Let's say $g\_2 = \xi \circ h\_2$ and because of $g\_1 = \xi \circ h\_1$ holds $g\_1 \prec g\_2$. Fortunately, this contradicts with compactness of space of measurable functions with finite codomain.


