Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $(X,M,\mu)$ be a measure and let $\{f_n\}$ be a sequence that is both uniformly integrable and tight on $X$. Suppose $f_n\to f$ a.e. and $f$ is finite a.e. How can we show that $f$ is integrable on $X$.

My approach is to write $X$ as $E_1\cup(X\setminus E_0)\cup (E_0\setminus E_1) $ . Then apply Fatou's lemma to conclude show that $f$ is integrable on $X\setminus E_1$. Since I have that $\mu(E_0)\lt \infty$ I can apply Fatou's lemma and Egorov, to show that $f$ is integrable on $E_1$ and $E_0\setminus E_1$.

Does this seem right?

share|improve this question

1 Answer 1

up vote 2 down vote accepted

I'm not sure what your sets are, but it seems you have the right idea. Here is an outline of one proof:

First prove the

Lemma: If $f_n\rightarrow f$ and $\int_E|f_n|\le M$ for each $n$, then $\int_E|f|\le M$. (Use Fatou.)

Having done that, proceed to prove the main result as follows:

Step 1) Use tightness to find a set $A$ of finite measure such that $\int_{A^C} |f_n|\le1$ for each $n$. Note by the Lemma then, that $\int_{A^C} |f|\le 1$.

Step 2) Use uniform integrability to find a $\delta>0$ such that whenever $\mu(E)<\delta$ then $\int_E |f_n|\le 1$ for each $n$. By the Lemma, then, we would have $\int_E |f|\le1$ for such $E$.

Step 3) Apply Egoroff to find $B\subset A$ with $\mu(A\setminus B)<\delta$ such that $f_n$ converges uniformly on $B$. Now pick $N$, so that for $m,n\ge N$, $\int_B|f_n-f_m|\le1$.

Step 4) From step 3) we have $\int_{ B}|f_n|\le 1+\int_X|f_N|$ for each $n\ge N$. Thus, there exists an $M$ so that $\int_{ B}|f_n|\le M$ for all $n$. Use the Lemma once more to conclude that $\int_{ B}|f|\le M$.

Step 5) Conclude that $\int_X |f| =\int_{A^C}|f|+\int_{A\setminus B}|f|+\int_B|f|<\infty$.

share|improve this answer
    
Thank you very much. What other ways exist of proving it? –  Jack Mar 27 '12 at 23:28
    
@Jack This is the only method that I'm aware of. But, try Googling "Vitali Convergence Theorem". –  David Mitra Mar 27 '12 at 23:31
    
Thanks. I'll do that. –  Jack Mar 27 '12 at 23:35
    
One more thing. How is the finiteness of $f$ used in the poof? –  Jack Mar 27 '12 at 23:47
    
@Jack It's needed for Egorov's Theorem. –  David Mitra Mar 27 '12 at 23:58

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.