Let $p$ and $q$ be finite Holder conjugates, and let $f_n$ be a sequence of $p$-integrable functions converging pointwise to $f$. Suppose there is some uniform constant $C$ such that $\|f_n\|_p < C$ for all $n$. Prove that if $g \in L^q$, then $$ \int f_ng \to \int fg. $$ The "obvious" estimate where one uses Holder's inequalty does not work here since we do not have $f \in L^p$ or that $\|f_n - f\||_p \to 0$. I can't find a dominating function, so I do not know how to use the convergence a.e. criterion.
Asked
Active
Viewed 969 times
1 Answers
3
Suppose $f_n \in L^p(X)$ and $f_n\rightarrow f$ a.e. then apply Fatou's lemma to the sequence $|f_n|^p$, which converges pointwise to $|f|^p$, then we see \begin{align} \int_X |f|^p\ d\mu \leq \liminf_{n\rightarrow \infty} \int_X |f_n|^p \leq C^p. \end{align} Hence $f \in L^p(X)$. To show $f_n$ converges weakly to $f$ in $L^p$, it suffices to show for all characteristic function $\chi_E$ where $\mu(E)<\infty$ we have \begin{align} \int_X f_n\chi_{E}\ d\mu =\int_E f_n\ d\mu\rightarrow \int_E f\ d\mu. \end{align}
Now, let us take out the big guns. Since $\|f_n\|_p \leq M$, then $f_n$ are uniformly integrable. Hence by Vitali Convergence Theorem, we indeed have \begin{align} \int_E f_n\ d\mu \rightarrow \int_E f\ d\mu. \end{align}
Edit: To prove uniform integrability, observe for every $A \subset E$ we have \begin{align} \int_A f_n\ d\mu \leq \| f_n\|_p \mu(A)^{1/q} \leq C\mu(A)^{1/q}. \end{align}
Jacky Chong
- 25,739
-
Why does the uniform bound imply uniform integrability? – Schmidt Oct 27 '16 at 07:39
-
Use Holder's inequality. – Jacky Chong Oct 27 '16 at 07:47
-
Ah right I forgot you reduced it to the case where the space is finite. – Schmidt Oct 27 '16 at 07:50
-
I can do that because $p , q \neq \infty$. – Jacky Chong Oct 27 '16 at 07:51
-
We are using $\sigma$-finiteness here, yes? – Schmidt Oct 27 '16 at 07:55
-
No. We used the fact that simple functions are dense in $L^p$ for $1\leq p <\infty$. – Jacky Chong Oct 27 '16 at 07:59