# $\{f_n\} \rightharpoonup f$ weakly in $L^P[a, b]$ iff; $\lim_{n\to \infty}\int_a^x f_n = \int_a^x f$

Let $$[a, b]$$ be a closed, bounded interval and $$1 < p < \infty$$. Suppose $$\{f_n\}$$ is a bounded sequence in $$L^P[a, b]$$ and f belongs to $$L^P[a, b]$$. Then $$\{f_n\} \rightharpoonup f$$ weakly in $$L^P[a, b]$$ if and only if; $$\lim_{n\to \infty}\int_a^x f_n = \int_a^x f$$

This is Thm 11 in ch.8 Royden and Fitzpatrick, but has no proof. Is there any proof in other books I can look at?

At least the only if direction is clear, for $$\chi_{[a,x]}$$ is in $$L^{q}$$, where $$q$$ is the conjugate of $$p$$.
For the other direction, note that it is not hard to see that $$\displaystyle\int_{a}^{b}f_{n}\varphi=\int_{a}^{b}f\varphi$$ for all step functions $$\varphi$$. The result will follow by the density of step functions in $$L^{q}$$.