I have a problem with showing the indirect implication

Asked Oct 22 '21 at 02:32

Active Oct 22 '21 at 02:32

Viewed 19 times

Let $(E,A,\mu)$ be a measure space
And let $(f_n)_{n \in IN}$ be a sequence of integrable functions that converges to an integrable function f.

Show that :

$\int\lvert f_n - f \lvert d\mu$ $\to$ 0 $\Leftrightarrow$ $\int \lvert f_n \lvert d\mu \to \int \lvert f \lvert d\mu$

(The integrals are over E)

asked Oct 22 '21 at 02:32

ML_john

2

One direction follows from Lipschitzness of $|\cdot|$. You need to show some work, or do a slightly better search or your question will get closed. – copper.hat Oct 22 '21 at 02:35
Yes I know , one implication is clear... , but the other one are very complicated. – ML_john Oct 22 '21 at 02:36
1

Maybe look up the generalised dominated convergence theorem. – copper.hat Oct 22 '21 at 02:39
1

It is not obvious, but not very complicated. – copper.hat Oct 22 '21 at 02:39
There is an other way too : If ${f_n / n \in IN}$ is an equi-integrable set, then there is a way to the proof – ML_john Oct 22 '21 at 02:43
1

https://math.stackexchange.com/q/72174/27978 – copper.hat Oct 22 '21 at 02:45
Thank you . – ML_john Oct 22 '21 at 02:57

0 Answers0