query: uniformly integrable family of functions equivalent statement
positive_passages.docid: 18942
positive_passages.text: An equivalent definition of uniform integrability
negative_passages.docid: 54008
negative_passages.text: The equivalence between Cauchy integral and Riemann integral for bounded functions
negative_passages.docid: 77322
negative_passages.text: Uniform continuity of $\sin(x\sin x )$ for all $x\in (0,\infty)$
negative_passages.docid: 123981
negative_passages.text: Is it possible to write any bounded continuous function as a uniform limit of smooth functions
negative_passages.docid: 128835
negative_passages.text: Compactness of the set of $n \times n$ orthogonal matrices
negative_passages.docid: 26566
negative_passages.text: Composition of two Riemann integrable functions
negative_passages.docid: 124061
negative_passages.text: Prove the reciprocal of a function is integrable if it is bounded
negative_passages.docid: 35547
negative_passages.text: Does $\{f_ng_n\}\to fg$ uniformly?
negative_passages.docid: 150305
negative_passages.text: $f_{n+1}(x):= \int_a ^x f_n(t)dt$, $\sum_{m=1} ^{\infty} f_m(x)$ is uniformly convergent
negative_passages.docid: 140936
negative_passages.text: If two integrals are equal, then the functions are the same
negative_passages.docid: 36129
negative_passages.text: Equivalence of convergence of a series and convergence of an infinite product
negative_passages.docid: 57609
negative_passages.text: Hahn-Banach proof by extension of basis
negative_passages.docid: 87119
negative_passages.text: A positive "Fourier transform" is integrable
negative_passages.docid: 63277
negative_passages.text: Gordan’s lemma equivalent statement
negative_passages.docid: 146839
negative_passages.text: Orthogonal Projection onto the $ {L}_{\infty} $ Unit Ball
negative_passages.docid: 139622
negative_passages.text: Generator of rational functions unchanged under $\sigma(X) = X + 1$
negative_passages.docid: 3856
negative_passages.text: Uniform Continuity of $x \sin x$