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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?

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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 $L^{q}$.

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