Let $(E,\mathcal {A}, \mu ) $ be a finite measure space and $X$ be a Banach space. The set of all Bochner-integrable functions from $E$ into $X$ is denoted by $\mathcal{L}_X^1$.
If $X$ is reflexive, we have the following theorem
Theorem 1
Let $ (f_n)_{n\geq 1} \subset \mathcal {L}_{X}^1$ is a sequence with : $$\sup_n \int_{E}{\|f_n\| d\mu} < \infty .$$ Then there exist $ h _{\infty} \in \mathcal {L}_{\mathbb {R}}^1 $ and a sub-sequence $ (g_k)_k $ of $(f_n)_n $ such that for every sub-sequence $ (h_m)_m $ of $(g_k)_k$ : $$ \frac{1}{i}\sum_{j=1}^{i}{h_j(t)}\to h _{\infty}(t) \text{ weakly in }X\text{ a.e. }$$
Proof of this result exists in the article "Infinite-dimensional extension of a theorem of Komlos" by Erik J. Balder (Theorem A).
If $X$ is not reflexive, we have the following theorem
Theorem 2
Let $ (f_n)_{n\geq 1} \subset \mathcal {L}_{X}^1$ is a sequence with : $$ \begin{cases} \bullet~~ \{f_n(t)\}\text{ is relatively weakly compact a.e.,}\\ \bullet~~ \sup_n \int_{E}{\|f_n\| d\mu} < \infty.\\ \end{cases} $$ Then there exist $ h _{\infty} \in \mathcal {L}_{\mathbb {R}}^1 $ and a sub-sequence $ (g_k)_k $ of $(f_n)_n $ such that for every sub-sequence $ (h_m)_m $ of $(g_k)_k$ : $$ \frac{1}{i}\sum_{j=1}^{i}{h_j(t)}\to h _{\infty}(t) \text{ weakly in }X\text{ a.e. }$$
Proof of this result exists in the article "Infinite-dimensional extension of a theorem of Komlos" by Erik J. Balder (Theorem B).
My problem:
I want an example of a non-reflexive Banach space and two sequences, such that:
The first sequence is bounded in $\mathcal {L}_{X}^1$ but it does not verify the consequence of Theorem 1.
The second sequence verifying the hypotheses of Theorem 2 and its consequences.
