# Show that for each $g \in L^q([0,1])$, $\lim_{n \to \infty} \int_0^1 f_ng = \int_0^1 fg$.

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.

• Though there is an answer here I decided to post an answer because there is a mistake in the counter-example in the link: math.stackexchange.com/questions/86893/… Dec 30, 2020 at 5:24

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.