Problem 15.10 of Real analysis for graduate students by R. Bass

Question

Suppose $1<p<\infty$ and q is conjugate exponent to p. Suppose $f_{n} \rightarrow f$ a.e where $sup_{n} ||f_{n}||_{p} < \infty$. If $g \in L^{q}$, then prove that $$Lim_{n\to \infty} \int f_{n}g =\int fg$$

Does this extend to the case when $f_{n} \rightarrow f$ a.e where, p=1 and $q=\infty$.

My attempt: I have the following $$|\int f_{n}g- \int{fg}\hspace{0.25cm} |=|\int (f_{n}g-fg)\hspace{0.25cm} | $$ $$=|\int (f_{n}-f)g\hspace{0.25cm} |\leq \int |(f_{n}-f)g| \leq ||f_{n}-f||_{p}||g||_{q} $$ By Holder's inequality, but now I suppose , that I can show that $||f_{n}-f||_{p} \to 0$, which will do the job. But can't show that.

I have also found the counter example.Consider $L^{1}[0,1]$. Let $f_{n}= n^{2}\chi_{[0,n^{-1}]}$, and $g=\chi_{[0,1]}$, then $g \in L^{\infty}([0,1])$ and $f_{n} \in L^{1}([0,1])$. Also $f_{n} \to 0$. This example I think perfectly works.

Thanks in advance for any help!!

Answer 1

You can proceed in this way (it's only a sketch of the proof).

1. Lemma 1: If $f_n\rightharpoonup h$ (meaning that $\int f_n g \to \int h g$ for every $g\in L^q$) and $f_n \to f$ a.e., then $h = f$ a.e.

2. Since $\|f_n\|_p \leq C$ for every $n$, then we can extract a subsequence $(f_{n_j})$ weakly convergent to some function $h \in L^p$.

3. From Lemma 1, we have $h=f$ a.e., hence by uniqueness of the limit the whole sequence $(f_n)$ weakly converges to $f$.

Answer 2

If the underlying measure ($\mu$) is finite, then we may use Egoroff's theorem. Let $h_n\equiv(f_n-f)g$ and fix $\eta>0$. Then we can pick $\epsilon>0$ s.t. if $\mu(B)<\epsilon$, then $\lVert g\times1_B\rVert_q< \eta$. Using Egoroff's theorem there exists $A_{\epsilon}$ s.t. $\mu(A_{\epsilon})<\epsilon$ and $f_n$ converges to $f$ uniformly on $A_{\epsilon}^c$. Then for all $n$ sufficiently large $|f_n-f|\le \eta$ on $A_{\epsilon}^c$ and $$ \int|h_n|=\int_{A_{\epsilon}}|h_n|+\int_{A_{\epsilon}^c}|h_n| \\ \le 2\sup_{n}\lVert f_n \rVert_p\times\lVert g\times1_{A_{\epsilon}}\rVert_q+\mu(A_{\epsilon}^c)\lVert g\rVert_q\times\eta\le C\eta $$ because $\lVert f_n-f\rVert_p\le 2\sup_n\lVert f_n\rVert_p$ by Minkowski inequality and Fatou's lemma.