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I have the following problem. There is a binary relation $\succeq$ defined on a separable metric space $X$, and $\succeq$ is total, transitive, and closed (as a subset of $X\times X$, in the product metric). There exists a set $Y\subseteq X$ that bounds the elements of $X$ in order (that is, for each $x \in X$ one can find $y_{1},y_{2} \in Y$ such that $y_{1}\succeq x\succeq y_{2}$). The set $X$ is itself convex, and the order $\succeq$ is affine when restricted to $Y$ (you may interpret the restriction of $\succeq$ to $Y$ as a convex subset of $Y \times Y$).

The equivalence class $[x]$ the element $x$ belongs to intersects the set $Y$. Informally, I want to construct a measurable selection from that intersection: a (Borel) measurable function that maps $x$ to an element $y$ of $Y$ such that $y\in [x]$. When the set $X$ is compact I can construct it easily. In general, for $X$ separable, I am not able to find one. I came across some measurable selection theorems, but they all work for Polish spaces. I also found a theorem in Kechris's book (Classical Descriptive Set Theory) that could be useful, but it also works for Polish spaces only.

My question is: Where can I find (measurable) selection theorems from equivalence classes? Is there any special theory for it (or counterexamples)? (Sorry for the vague question, but I am not quite acquainted with the literature.)

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Some books that might help:

A Course in Borel Sets by S.M. Srivastava

Selectors by J.E. Jayne and C.A. Rogers

Also, as a suggestion, if you tag your questions better (this one with say 'topology' or 'set theory'), it'll help you get better, quicker answers from the experts here.

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