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Hey, I was just wondering, I'm using some of Robert Aumann's ideas about measurable structures on function spaces (From his paper 'Borel structures for Function spaces': http://projecteuclid.org/euclid.ijm/1255631584) and I had a question. Let $(X,\Sigma_{X})$, $(Y,\Sigma_{Y})$ be measurable spaces, let $Y^{X}$ be the space of measurable functions from $X$ to $Y$, I want to define a measurable structure on a subset $F$ of $Y^{X}$, using his construction I managed to define a measurable structure on $F$, now the thing here is that I need that structure to be discrete for some stuff I'm working on. I was thinking and couldn't find a way around it so I remembered that $F$ is actually the set of continuous functions from $X$ to $Y$ (which are of course measurable), so I was thinking, what if I simply use the function space $Y_{C}^{X}$ (the space of continuous functions from $X$ to $Y$) and equip it with the discrete topology? I was thinking that since it is a topological space I can define a measurable space generated by the open sets (which are all the subsets) on $Y_{C}^{X}$, denoted by $(Y_{C}^{X},\Sigma_{Y_{C}^{X}})$, which would have a discrete measurable structure, but I'm not sure about doing this, so my question is:

If I take the set of continuous functions $Y_{C}^{X}$ between 2 topological spaces $X$ and $Y$ and define a topology on it (in this case the discrete topology), can I then define a Borel structure on it generated by the open sets? Because then I'd have the discrete measurable space I'm looking for, this seems like the logical thing to do but I don't know if I'll running run into some conceptual problems if I do this, I don't know if this has been done as Aumann doesn't mention it in his paper

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Hey, I was just wondering, I'm using some of Robert Aumann's ideas about measurable structures on function spaces (From his paper 'Borel structures for Function spaces': http://projecteuclid.org/euclid.ijm/1255631584) and I had a question. Let $(X,\Sigma_{X})$, $(Y,\Sigma_{Y})$ be measurable spaces, let $Y^{X}$ be the space of measurable functions from $X$ to $Y$, I want to define a measurable structure on a subset $F$ of $Y^{X}$, using his construction I managed to define a measurable structure on $F$, now the thing here is that I need that structure to be discrete for some stuff I'm working on. I was thinking and couldn't find a way around it so I remembered that $F$ is actually the set of continuous functions from $X$ to $Y$ (which are of course measurable), so I was thinking, what if I simply use the function space $Y_{C}^{X}$ (the space of continuous functions from $X$ to $Y$) and equip it with the discrete topology? I was thinking that then since it is a topological space I can define a measurable space generated by the open sets (which are all the subsets) on $Y_{C}^{X}$, denoted by $(Y_{C}^{X},\Sigma_{Y_{C}^{X}})$, which would have a discrete measurable structure, but I'm not sure about doing this, so my question is:

If I take the set of continuous functions $Y_{C}^{X}$ between 2 topological spaces $X$ and $Y$ and define a topology on it (in this case the discrete topology), can I then define a Borel structure on it generated by the open sets? Because then I'd have the discrete measurable space I'm looking for, this seems like the logical thing to do but I don't know if I'll running into some conceptual problems if I do this, I don't know if this has been done as Aumann doesn't mention it in his paper

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# A question about measurable structures on function spaces

Hey, I was just wondering, I'm using some of Robert Aumann's ideas about measurable structures on function spaces (From his paper 'Borel structures for Function spaces': http://projecteuclid.org/euclid.ijm/1255631584) and I had a question. Let $(X,\Sigma_{X})$, $(Y,\Sigma_{Y})$ be measurable spaces, let $Y^{X}$ be the space of measurable functions from $X$ to $Y$, I want to define a measurable structure on a subset $F$ of $Y^{X}$, using his construction I managed to define a measurable structure on $F$, now the thing here is that I need that structure to be discrete for some stuff I'm working on. I was thinking and couldn't find a way around it so I remembered that $F$ is actually the set of continuous functions from $X$ to $Y$ (which are of course measurable), so I was thinking, what if I simply use the function space $Y_{C}^{X}$ (the space of continuous functions from $X$ to $Y$) and equip it with the discrete topology? I was thinking that then since it is a topological space I can define a measurable space generated by the open sets (which are all the subsets) on $Y_{C}^{X}$, denoted by $(Y_{C}^{X},\Sigma_{Y_{C}^{X}})$, which would have a discrete measurable structure, but I'm not sure about doing this, so my question is:

If I take the set of continuous functions $Y_{C}^{X}$ between 2 topological spaces $X$ and $Y$ and define a topology on it (in this case the discrete topology), can I then define a Borel structure on it generated by the open sets? Because then I'd have the discrete measurable space I'm looking for, this seems like the logical thing to do but I don't know if I'll running into some conceptual problems if I do this, I don't know if this has been done as Aumann doesn't mention it in his paper