Question

Why does L^1 convergence follow from uniform convergence for finite measure spaces? I think I have an idea of how to show it, but I'd like to discuss it.

Answer 1

err what is this in relation to specifically what topic?

Answer 2

Real analysis, measure theory, etc.

Answer 3

is L a random variable??

Answer 4

L^1 is the space of functions that satisfy the following condition:
\(\int |f| < \infty\)

Answer 5

ahh sorry this is beyond my scope...apologies

Answer 6

@Princer_Jones perhaps might be able to help u

Answer 7

So, let's start with a finite measure space \((X, \mathcal{M}, \mu)\). In particular this means that \(\mu(X) < \infty\). We want to show that for any sequence of functions \(\{f_n \}_1^\infty\subset L^1(\mu)\) which converges uniformly \(f_n \to f\) to some function \(f\). That we also have \(L^1(\mu)\) convergence and by that I mean \(f \in L^1(\mu)\) and \(\int f_n \to \int f \ \)

Answer 8

My idea is to consider \(f_n - f\) and apply the dominated convergence theorem. Since we have uniform convergence for every \(\varepsilon > 0\) we can certainly find some \(N\) such that for all \(n > N\) we have \(|f_n - f| < \epsilon\). Then, can't we apply the dominated convergence theorem since the constant function \(g(x) = \epsilon\) is surly \(L^1\) for finite measure spaces.

Answer 9

*

Answer 10

Of course the argument fails (as it should) for non-finite measure spaces since a constant function would have an infinite integral.

Answer 11

yes this seems the correct way,any sequence {f_n(z)} will be uniform convergent ,if for every given positive numbet (epsilon)>0 ,and for all Z in D
and there should be an \[N \in IN~exist~which~is~independent~of~Z~such~that\\|f_n(Z)-f_m(Z)|<\epsilon,\forall n,m \ge N\]

Answer 12

it is cauchys criteria of uniform convergence i think

Answer 13

and this definition also holds when {f_n(x)} is a sequence of real function in a space like L^1

Answer 14

is this a million dollar problem ? :P

Answer 15

No, it is just a problem in Folland Real Analysis: http://www.amazon.ca/Real-Analysis-Modern-Techniques-Applications/dp/0471317160