Measures Integrals And Martingales 2nd Edition René L. Schilling - Solutions
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![Theorem 28.6 (Weierstra) Polynomials are dense in C[0, 1] w.r.t. uniform convergence. Proof (S. N. Bernstein)](https://dsd5zvtm8ll6.cloudfront.net/images/question_images/1705/9/8/7/29665af4ce02bec11705987295385.jpg)
![since the function p-p(1-p) attains its maximum at p=1/2. As u C[0, 1] is uniformly continuous, u(x) - u(y) |](https://dsd5zvtm8ll6.cloudfront.net/images/question_images/1705/9/8/7/30865af4cec958951705987308031.jpg)
![Corollary 28.8 The monomials (t")neN, are complete in L= L ([0, 1], dt), that is, 0,11u(t)t" dt=0 for all](https://dsd5zvtm8ll6.cloudfront.net/images/question_images/1705/9/8/7/33865af4d0a60c5f1705987337820.jpg)
![Theorem 21.5 (Riesz) Let (X,,u) be a o-finite measure space, p = [1,00) and qe (1,00] conjugate indices.](https://dsd5zvtm8ll6.cloudfront.net/images/question_images/1705/9/1/3/97265ae2e74285a01705913971457.jpg)
![Theorem 21.5 (Riesz) Let (X, ,) be a o-finite measure space, p = [1,00) and qe (1,00] conjugate indices.](https://dsd5zvtm8ll6.cloudfront.net/images/question_images/1705/9/1/3/86665ae2e0a79cf71705913866010.jpg)
![Lemma 24.1 (Doob's upcrossing estimate) Let (un)neN be a submartingale and denote by U([a, b]; N; x) the](https://dsd5zvtm8ll6.cloudfront.net/images/question_images/1705/9/2/3/79865ae54d66319d1705923797527.jpg)
![the restricted maximal function. Since (u-on) 3] 3} } + X"{x:on(x) - u(x) > } Cn "||un||1+0+||on-u||1, where](https://dsd5zvtm8ll6.cloudfront.net/images/question_images/1705/9/2/5/93965ae5d33345871705925938205.jpg)
