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accepted A certain family of continuous functions on $[0,1]^2$ the closure of which linear span is $\tilde{\mathcal{C}}([0,1]^2,\mathbb{R}))$
Jul
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accepted Approximation of bounded measurable functions with continuous functions
Jul
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comment A certain family of continuous functions on $[0,1]^2$ the closure of which linear span is $\tilde{\mathcal{C}}([0,1]^2,\mathbb{R}))$
@ Davide: yes this is actually what i wanted ^^
Jul
30
comment A certain family of continuous functions on $[0,1]^2$ the closure of which linear span is $\tilde{\mathcal{C}}([0,1]^2,\mathbb{R}))$
I am heading over to the "why didn't I see it myself board" to bang my head against it - will be back in a couple of minutes :D
Jul
30
comment A certain family of continuous functions on $[0,1]^2$ the closure of which linear span is $\tilde{\mathcal{C}}([0,1]^2,\mathbb{R}))$
Thank you Davide - I edited the question. In it's initial form it was unlcear. I actually meant the space $\tilde{\mathcal{C}}([0,1]^2,\mathbb{R}))$ (as described above) and not $\mathcal{C}([0,1]^2,\mathbb{R}))$ like initially written. As I see it $(1-x)(1-y)f(x,y)$ seems already to answer my question. Just write you comment as an answer so that I can accept it :)
Jul
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revised A certain family of continuous functions on $[0,1]^2$ the closure of which linear span is $\tilde{\mathcal{C}}([0,1]^2,\mathbb{R}))$
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30
awarded  Commentator
Jul
30
revised A certain family of continuous functions on $[0,1]^2$ the closure of which linear span is $\tilde{\mathcal{C}}([0,1]^2,\mathbb{R}))$
added 228 characters in body
Jul
30
asked A certain family of continuous functions on $[0,1]^2$ the closure of which linear span is $\tilde{\mathcal{C}}([0,1]^2,\mathbb{R}))$
Jul
30
comment Approximation of bounded measurable functions with continuous functions
I assume $(a,b)$ is meant to be an open square in $\R^2$ ?
Jul
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comment Approximation of bounded measurable functions with continuous functions
Okey - one can approximate any $1_{(a,b)}$ Function via continuous functions and then use the argument you have provided to show the statement for bounded functions. Isn't that basically the monotone class theorem (the argument seems pretty familiar to me)
Jul
29
comment Approximation of bounded measurable functions with continuous functions
@copper.hat The complaint wasn't directed at you - I was just frustrated. English is not my native languange and sometimes the questions might lack presicion at first. Before voting down users should at least say what is wrong and give the "poster" a chance to edit the question :) Again not directed at you just some general thoughts on the subject :D
Jul
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revised Approximation of bounded measurable functions with continuous functions
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awarded  Supporter
Jul
29
comment Approximation of bounded measurable functions with continuous functions
Okey I will try - it will be somewhat difficult, for my question emerged while reading one of the last proofs in the paper and there are lots of perliminary resuls flowing into it.
Jul
29
comment Approximation of bounded measurable functions with continuous functions
So basically if $K$ is compact and I am dealing with the Borel or Lebesgue-Mesure - the approach should work? Does this theorem have a name or do you by any chance know of any citeable reference.
Jul
29
comment Approximation of bounded measurable functions with continuous functions
the Montone Class Theorem is an approach I haven't tried. I am only familiar with it from the perspective of measure theory, where one uses indicator functions and Dynkin's Theorem. Thanks
Jul
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revised Approximation of bounded measurable functions with continuous functions
added 409 characters in body
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comment Approximation of bounded measurable functions with continuous functions
It is not a homework problem ... STOP VOTING DOWN POSTS in advance - really I hate that. I was reading a paper and the authors just wrote "the usual limiting argument provides the result for all bounded functions" - prior to that they showed it for continuous functions. So I was wondering what "the usual limiting argument might be" ... Therefore I do not even know what kind of approximation is needed - uniform is alsways nice but I think pointwise would also work.