Let $1 <p < \infty$. Consider a sequence $(f_n)_n \subset L^p(0,1)$ and let $f \in L^p(0,1)$. Prove that $f_n$ is weakly convergent to $f$ if and only if the sequence $(f_n)_n$ is bounded in $L^p(0,1)$ and $\forall x \in [0,1]$, $\int_{0}^{x} f_n (y) dy \rightarrow \int_{0}^{x} f (y) dy $ (*). Discuss if the equivalence still holds removing the hypothesis that $(f_n)_n$ is bounded in $L^p(0,1)$.
Concerning the first part of the exercise, I reasoned as follows.
The direct implication is straight-forward since every weakly convergent sequence in a normed space is bounded. Moreover, from the characterization of weak convergence in $L^p$ spaces, one has that $(f_n)_n$ converges weakly in $L^p(0,1)$ $\Leftrightarrow$ $\forall g \in L^q(0,1)$, $\int_{0}^{1} f_n(y) g(y) dy \rightarrow \int_{0}^{1} f(y) g(y) dy$ (where $q$ is such that $\frac{1}{p} + \frac{1}{q} = 1$). Thus, fixing an arbitrary $x \in [0,1]$, we can choose $g = \chi_{[0,x]} \in L^q(0,1)$ and, by weak convergence of $(f_n)_n$ we can conclude that $\int_{0}^{x} f_n (y) dy \rightarrow \int_{0}^{x} f (y) dy $.
The inverse implication is less obvious. My first idea was to use the dominated convergence theorem, but I cannot find a suitable domination for any arbitrary function $g \in L^q(0,1)$. I would like to use somehow the hypothesis regarding the integral of $f_n$, but I was not able to satisfyingly develop the reasoning. Therefore, I proceeded in the following way. I know that for $1<p<\infty$, $L^p(0,1)$ is reflexive. By hypothesis, the sequence $(f_n)_n$ is bounded in $L^p(0,1)$, thus there exists a subsequence $(f_{n_k})_k$ which converges weakly to a function $\overline{f} \in L^p(0,1)$. Being a subsequence of $(f_n)_n$, it is still true that $\forall x \in [0,1]$, $\int_{0}^{x} f_{n_k} (y) dy \rightarrow \int_{0}^{x} f (y) dy$ and, following from weak convergence, we also have that $\int_{0}^{x} f_{n_k} (y) dy \rightarrow \int_{0}^{x} \overline{f} (y) dy$. Thus, from the arbitrarity of $x \in [0,1]$ and from the uniqueness of the limit, we can conclude that $f = \overline{f}$. This reasoning can be iterated to a subsequence of $(f_{n_k})_k$ and thus the weak convergence of $f_n$ to $f$ can be deduced applying Urysohn lemma.
On the last part of the exercise I have few ideas. My guess is that the equivalence does not hold, hence I made an attempt to construct a sequence $(f_n)_n$ such that $||f_n||_{p} = n$ for all $n \in \mathbb{N}$. However, I cannot find one which satisfies the integral property (*).
If anyone could give me a hint on how to proceed (as well as a check on the reasoning above), It would be greatly appreciated.