# application on Lebesgue Dominated Convergence Theorem

If $$\{f_n\}$$ is a sequence of measurable functions on the measurable set $$E$$ such that $$f_n \rightarrow f$$ converges pointwise almost everywhere on $$E$$ where $$|f| \in L(E)$$ "i.e. $$|f|$$ is Lebesgue integrable on $$E$$, say, $$\int_{E} |f| = a$$ and $$\lim_{n \rightarrow \infty} \int_{E} |f_n(x)|= b$$.

I need to show that $$\lim_{n \rightarrow \infty} \int_{E} |f_n(x)-f(x)|$$ exists and to find its value. I can see that $$|f_n(x)-f(x)| \leq |f_n(x)|+|f(x)|$$ so passing the integral over $$E$$ then taking the limit as $$n \rightarrow \infty$$ proves that the limit of the integral exists, but I am still not sure how to find the limit value, my guess is to apply the Lebesgue Dominated Convergence Theorem or the general Lebesgue Dominated Convergence Theorem.

• Hmm. $(-1)^n\le 2$ and $\lim 2$ exists, hence $\lim (-1)^n$ exists? – David C. Ullrich Mar 23 at 21:43
• I passed the limit into positive terms, right? – math_for_ever Mar 23 at 22:30
• so $0<2+(-1)^n \le 4$ and $\lim 4$ exists, hence $\lim(2+(-1)^n)$ exists. got it, I didn't see how positivity mattered... – David C. Ullrich Mar 23 at 22:57

Here is a hint: apply Fatou's lemma to $$|f_n - f| + |f| - |f_n| \ge 0$$ to find that $$b-a \le \liminf \int |f_n - f|.$$ Then apply Fatou's lemma to $$|f_n| + |f| - |f_n - f|$$ to find in turn that $$\limsup \int |f_n - f| \le b-a.$$

• By the way this forces $a \le b$. – Umberto P. Mar 23 at 21:53
• Thank you ! that makes the problem more sense. – math_for_ever Mar 23 at 22:28

It seems like this, I am not sure if it is correct.

Please tell me where I was wrong.

Since $$\lim_{n}\int_E|f_n(x)|=b \lt ∞$$, $${\exists}$$ $$N$$ such that $${\forall} n \gt Nf_n\in L(E)$$.

Since $$f_n \rightarrow f a.e.$$, we have $$f_n - f \rightarrow 0 a.e.$$.

And $$|f_n - f|$$ domined by $$|f_n| + |f|$$, by Lebesgue Dominated Convergence Theorem $$\int_E|f_n - f| \rightarrow \int_E 0 =0$$.