Talk:Weierstrass Approximation Theorem/Proof 2

Probabilistic structure
I am not very familiar with probabilistic proofs. Are you sure that such a proof does not require an assumption (trivial, natural, canonical, induced etc.) about the probability space you use? Maybe in the long term this proof should have its own page rather than being a second proof?--Julius (talk) 21:18, 17 October 2022 (UTC)


 * It's a proof of Weierstrass Approximation Theorem, and it's already got its own page, I don't udnerstand. --prime mover (talk) 21:21, 17 October 2022 (UTC)


 * This is quite a common proof of this theorem. I think $X_k \sim \Binomial 1 p$ is enough for everything else to work. Caliburn (talk) 21:32, 17 October 2022 (UTC)


 * Random variables need a probability space which is an additional structure on top of my assumptions (its like comparing proofs in topological, metric and normer vector spaces - analogous but not congruent). Now I see the addition of Kolmogorov extension theorem which is supposed to remedy this issue. I wonder whether this theorem says that functions spaces belong to probabilistic spaces like metric spaces belong to topological spaces. Then I would count such proofs analogous but not equal. Congruent proofs would stick to function spaces but involve different lemmas.--Julius (talk) 21:43, 17 October 2022 (UTC)

Probabilistic structure
I am not very familiar with probabilistic proofs. Are you sure that such a proof does not require an assumption (trivial, natural, canonical, induced etc.) about the probability space you use? Maybe in the long term this proof should have its own page rather than being a second proof?--Julius (talk) 21:18, 17 October 2022 (UTC)


 * It's a proof of Weierstrass Approximation Theorem, and it's already got its own page, I don't udnerstand. --prime mover (talk) 21:21, 17 October 2022 (UTC)


 * This is quite a common proof of this theorem. I think $X_k \sim \Binomial 1 p$ is enough for everything else to work. Caliburn (talk) 21:32, 17 October 2022 (UTC)


 * I believe I understand your concern about the assumption. I added a remark on Kolmogorov Extension Theorem. We can also give a space explicitly as $\Omega := \set {0,1}^\N$ and $\Pr := \paren { \paren{1-p} \delta_0 + p \delta_1 }^{\otimes \N}$.
 * The proof has a solid justification but it is currently hard to add more details because Category:Probability Theory is not sufficiently developed yet. --Usagiop (talk) 21:49, 17 October 2022 (UTC)