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Let $1 < p < \infty$. Let $\{f_n\}$ be a sequence of functions in $L^p([0,1])$ that converges almost everywhere to a function $f \in L^p([0,1])$. Also, suppose there exists a constant $M$ such that $\|f_n\|_p \leq M$ for all $n$.

Show that for each $g \in L^q([0,1])$, where $\frac{1}{p} + \frac{1}{q} = 1$, we have

$$ \lim_{n \to \infty} \int_0^1 f_ng = \int fg.$$

This involves either BCT or DCT and Holder's inequality somewhere, but we cannot figure out how to start. We can bound $\|f_ng\|_1 \leq M\|g\|_q$ by Holder's Inequality, but after that we are stuck. That fact may not even be useful.

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First consider a bounded measurable function $g$. Then the conclusion follows from the fact that $\int |f_n-f| \to 0$. This well known fact is proved using uniform integrability of the sequence $(f_n)$. [See below for a proof].

The general case is proved easily using Holder's inequlaity: There exist a bounded measurable function $h$ such that $\int |g-h|^{q} <\epsilon$. Note that $|\int f_ng -\int f_nh | \leq \|f_n\|_p \|g-h\|_2 \leq M \|g-h\|_2$ and $|\int fg -\int fh | \leq \|f\|_p \|g-h\|_2 \leq M \|g-h\|_2$.

Let $f_n \to f$ a.e. on $[0,1]$ and assume that $\|f_n\|_p $ is bounded for some $p \in (1,\infty)$. Then $\int |f_n-f| \to 0$.

Proof: It is clearly enough to prove this when $f=0$ (since $\|f\|_p$ is necessarily finite by Fatou's Lemma).

Now $$m\{x: |f_n(x) |>C\}) \leq \frac 1 {C^{p}} \int |f_n|^{p}.$$

From Holder's inequality we get $$\int_{\{x: |f_n(x) |>C\}} |f_n| \leq \|f_n\|_p (m\{x: |f_n(x) |>C\})^{1/q}.$$ It follows that $\int_{\{x: |f_n(x) |>C\}} |f_n|$ can be made less than $\epsilon$ for all $n$ by choosing $C$ large enough. On the other hand $\int_{\{x: |f_n(x) |\leq C\}} |f_n| \to 0$ by DCT.

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