12 нояб. 2012 г. · Uniform convergence implies uniform boundedness assuming each fn is a bounded function. For example, consider the sequence fn:(0,1)→R given by f ...

8 нояб. 2015 г. · The assumption of the statement is that fn and f are point-wise bounded by some function g and that g is integrable.

17 мар. 2015 г. · If (fn) is a pointwise convergent, uniformly bounded sequence of measurable functions on I, then its limit f is also measurable.

12 мар. 2017 г. · I am searching for a counter example to the fact that bounded convergence theorem holds if we drop the assumption that sequence is uniformly bounded.

17 июл. 2016 г. · Theorem 1.4 (Bounded convergence theorem) Suppose that {fn} is a sequence of measurable functions that are all bounded by M, are supported on a ...

17 дек. 2012 г. · In order to bound the integral of a function, we need to bound either the measure of the domain of the integral, or the function itself. By ...

5 июн. 2012 г. · Theorem: Let {fn} be a sequence of functions bounded by M, supported on a set E of finite measure, and fn(x)→f(x) a.e. x. Then f is bounded ...

22 апр. 2021 г. · I'm trying to understand how these convergence theorems are induced from uniform convergence, Egorov's theorem, and Fatou's lemma.

23 мар. 2021 г. · I think I am very close. To satisfy the BND convergence theorem, we need the input to be of finite measure, which we have: m([0,1])=1−0=1<∞. We ...

2 авг. 2016 г. · Theorem 13-17(Arzelˊa). Assume that {fn} is a uniformly bounded, pointwise convergent on [a,b] and suppose that each fn is Riemann-integrable ...