Recent content by bbkrsen585

But, how do we know that |gn|^p converges to |g|^p a.e.? If gn goes to g almost everywhere does that imply that |gn|^p converges to |g|^p a.e.? That's not so clear to me. Thanks.

Homework Statement Hi guys. I have one question regarding convergence in Lp. Suppose gn converges to g mu-almost everywhere on [0,1]. Suppose further that \left\|gn\left\|p\rightarrowM < \infty. How do I show that the pointwise limit g is in Lp? Thanks. Homework Equations So far, I...

Yes, I agree with your argument, but I'm just not seeing how this shows uniform convergence. LDCT gives us that the integral of gn converges point-wise to the integral of g on [0,1]. How can we infer from this that Gn(x) goes uniformly to Gn? Maybe what I'm not seeing is that we have...

Homework Statement Hi All, I've been having great difficulty making progress on this problem. Suppose gn converges to g a.e. on [0,1]. And, for all n, gn and h are integrable over [0,1]. And |gn|\leqh for all n. Define Gn(x)=\intgn(x) from 0 to x. Define G(x)=\intg(x) from 0 to x...

Suppose gn are nonnegative and integrable on [0, 1], and that gn \rightarrow g almost everywhere. Further suppose that for all \epsilon > 0, \exists \delta > 0 such that for all A \subset [0, 1], we have meas(A) < \delta implies that supn \intA |gn| < \epsilon. Prove that g is integrable...

Hi there. I was wondering if someone could help me out with the following. Let E1, E2, ... be a sequence of nondecreasing measurable sets, each with finite measure. Define E = \bigcupEk, where E is the union of an infinite number of sets Ek. Suppose f is measurable and Lebesgue integrable...

I want to prove the following proposition: Given any uncountable set of real numbers S, there exists a countable sub-collection of numbers in S, whose sum is infinite. Please point me in the right direction.

So I've thought about that, but here's the thing. Suppose I assume that B is uncountable. Then, for each y in B, there is a corresponding open image that has measure greater than zero. Since we have uncountably many y's (by assumption), there would be an uncountably many number of open images...

I want to prove the following. Statement: Given that f is measurable, let B = {y \in ℝ : μ{f^(-1)(y)} > 0}. I want to prove that B is a countable set. (to clarify the f^(-1)(y) is the inverse image of y; also μ stands for measure) Please set me in the right direction. I would greatly...