Let $X$ be a polish space equiped with the borel sigma-algebra and a probability measure $\mu$. How can one show that the set of all borel measurable functions $f:X\rightarrow R $ ($R$ being the real numbers), where two a.e. equal functions are identified, equiped with the topology of convegence in measure is separable?
How can one show that the topology of convergence in measure is separable?
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real-analysis
measure-theory
functional-analysis
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2I've given you *many* references in [my last answer](http://math.stackexchange.com/questions/58018/radon-nikodym-derivative-as-a-measurable-function-in-a-product-space) to you. Fisher-Witte Morris-Whyte (rem 2.4) give a detailed reference to Wheeden-Zygmund, and it is contained in at least three of the other references I've given to you. Please look around a bit before you ask. You can also see Proposition 7 [here](http://dx.doi.org/10.1090/S0002-9947-1976-0414775-X) for a more precise and general result (there polonais = Polish). – 2011-08-19
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0Thank you Theo, you have helped me greatly. I did look around, but didn't find it. I will look in some of your references... BTW I haven't found it in Wheeden-Zygmund – 2011-08-19
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0Hmmm. Not having one of my most charitable days, I guess... In fact, in the books I mentioned it's in the exercises (e.g. Kechris) and at times not even explicitly mentioned (e.g. in Wheeden-Zygmund), you're right. It's one of those results that people like to leave as an exercise... Sorry about that. Moore's proof of proposition 7 mentioned above suffers a bit from its generality. BTW: I flagged your account for merging with the other one. You could consider registering your account, then it is easier for the software to recognize you. – 2011-08-19
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0@Arnold: I have merged your other account, A_0, into your current account. If you have trouble logging in, or if you accidentally create duplicate accounts, simply flag one of your own questions for moderator attention, and we will help out. – 2011-08-19
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0Thanks (at least 15 characters...) – 2011-08-19