Signed measures and uniform integrability

[qpt]

Hello,

If u is a positive measure, I need to show that any finite subset of L^1(u) is uniformly integrable, and if {fn} is a sequence in L^1(u) that converges in the L^1 metric to f in L^1(u), then {fn} is uniformly integrable.

I know that a collection of functions {f_alpha}_alpha_in_A subset L^1(u) is uniformly integrable if for every e > 0 there exists a d > 0 such that |int_E f_alpha du| < epsilon for all alpha in A whenever u(E) < delta, but I am stuck on the proofs. Any assistance is appreciated, thanks.

 

[fresh_42]

These notes can possibly help:
Uniform Integrability (12 pages)
https://web.ma.utexas.edu/users/gordanz/notes/uniform_integrability.pdf