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 Nov5 awarded Popular Question Aug5 accepted A certain family of continuous functions on $[0,1]^2$ the closure of which linear span is $\tilde{\mathcal{C}}([0,1]^2,\mathbb{R}))$ Jul30 accepted Approximation of bounded measurable functions with continuous functions Jul30 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 ^^ Jul30 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 Jul30 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 :) Jul30 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 Jul30 awarded Commentator Jul30 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 Jul30 asked A certain family of continuous functions on $[0,1]^2$ the closure of which linear span is $\tilde{\mathcal{C}}([0,1]^2,\mathbb{R}))$ Jul30 comment Approximation of bounded measurable functions with continuous functions I assume $(a,b)$ is meant to be an open square in $\R^2$ ? Jul29 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) Jul29 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 Jul29 revised Approximation of bounded measurable functions with continuous functions added 947 characters in body Jul29 awarded Supporter Jul29 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. Jul29 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. Jul29 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 Jul29 revised Approximation of bounded measurable functions with continuous functions added 409 characters in body Jul29 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.